{"id":27850,"date":"2024-08-13T20:43:31","date_gmt":"2024-08-13T20:43:31","guid":{"rendered":"http:\/\/toposuranos.com\/material\/?p=27850"},"modified":"2024-09-12T22:08:59","modified_gmt":"2024-09-12T22:08:59","slug":"%d0%bf%d1%80%d0%b5%d0%b4%d0%b5%d0%bb-%d1%84%d1%83%d0%bd%d0%ba%d1%86%d0%b8%d0%b8-%d0%be%d0%b4%d0%bd%d0%be%d0%b9-%d0%b4%d0%b5%d0%b9%d1%81%d1%82%d0%b2%d0%b8%d1%82%d0%b5%d0%bb%d1%8c%d0%bd%d0%be%d0%b9","status":"publish","type":"post","link":"https:\/\/toposuranos.com\/material\/ru\/%d0%bf%d1%80%d0%b5%d0%b4%d0%b5%d0%bb-%d1%84%d1%83%d0%bd%d0%ba%d1%86%d0%b8%d0%b8-%d0%be%d0%b4%d0%bd%d0%be%d0%b9-%d0%b4%d0%b5%d0%b9%d1%81%d1%82%d0%b2%d0%b8%d1%82%d0%b5%d0%bb%d1%8c%d0%bd%d0%be%d0%b9\/","title":{"rendered":"\u041f\u0440\u0435\u0434\u0435\u043b \u0444\u0443\u043d\u043a\u0446\u0438\u0438 \u043e\u0434\u043d\u043e\u0439 \u0434\u0435\u0439\u0441\u0442\u0432\u0438\u0442\u0435\u043b\u044c\u043d\u043e\u0439 \u043f\u0435\u0440\u0435\u043c\u0435\u043d\u043d\u043e\u0439"},"content":{"rendered":"<p><center><\/p>\n<h1>\u041f\u0440\u0435\u0434\u0435\u043b \u0444\u0443\u043d\u043a\u0446\u0438\u0438 \u043e\u0434\u043d\u043e\u0439 \u0434\u0435\u0439\u0441\u0442\u0432\u0438\u0442\u0435\u043b\u044c\u043d\u043e\u0439 \u043f\u0435\u0440\u0435\u043c\u0435\u043d\u043d\u043e\u0439<\/h1>\n<p><\/center><\/p>\n<p style=\"text-align:center\"><em><strong>\u0420\u0435\u0437\u044e\u043c\u0435:<\/strong><br \/>\n\u041d\u0430 \u044d\u0442\u043e\u043c \u0437\u0430\u043d\u044f\u0442\u0438\u0438 \u043f\u043e\u0434\u0440\u043e\u0431\u043d\u043e \u0440\u0430\u0441\u0441\u043c\u0430\u0442\u0440\u0438\u0432\u0430\u0435\u0442\u0441\u044f \u0444\u043e\u0440\u043c\u0430\u043b\u044c\u043d\u043e\u0435 \u043e\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u0438\u0435 \u043f\u0440\u0435\u0434\u0435\u043b\u0430 \u0444\u0443\u043d\u043a\u0446\u0438\u0438 \u043e\u0434\u043d\u043e\u0439 \u0434\u0435\u0439\u0441\u0442\u0432\u0438\u0442\u0435\u043b\u044c\u043d\u043e\u0439 \u043f\u0435\u0440\u0435\u043c\u0435\u043d\u043d\u043e\u0439, \u0438 \u043d\u0430 \u043e\u0441\u043d\u043e\u0432\u0435 \u044d\u0442\u043e\u0433\u043e \u0434\u0435\u043c\u043e\u043d\u0441\u0442\u0440\u0438\u0440\u0443\u044e\u0442\u0441\u044f \u043e\u0441\u043d\u043e\u0432\u043d\u044b\u0435 \u0441\u0432\u043e\u0439\u0441\u0442\u0432\u0430, \u043a\u043e\u0442\u043e\u0440\u044b\u0435 \u043f\u0440\u0438\u0432\u043e\u0434\u044f\u0442 \u043a \u0430\u043b\u0433\u0435\u0431\u0440\u0435 \u043f\u0440\u0435\u0434\u0435\u043b\u043e\u0432.<\/br><\/em><\/p>\n<p style=\"text-align:center\"><em><strong>\u0426\u0435\u043b\u0438 \u043e\u0431\u0443\u0447\u0435\u043d\u0438\u044f:<\/strong><br \/>\n\u041a \u043a\u043e\u043d\u0446\u0443 \u044d\u0442\u043e\u0433\u043e \u0437\u0430\u043d\u044f\u0442\u0438\u044f \u0441\u0442\u0443\u0434\u0435\u043d\u0442 \u0441\u043c\u043e\u0436\u0435\u0442:<\/p>\n<ul>\n<li><strong>\u0417\u0430\u043f\u043e\u043c\u043d\u0438\u0442\u044c<\/strong> \u043e\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u0438\u0435 \u043f\u0440\u0435\u0434\u0435\u043b\u0430 \u0444\u0443\u043d\u043a\u0446\u0438\u0438 \u043e\u0434\u043d\u043e\u0439 \u0434\u0435\u0439\u0441\u0442\u0432\u0438\u0442\u0435\u043b\u044c\u043d\u043e\u0439 \u043f\u0435\u0440\u0435\u043c\u0435\u043d\u043d\u043e\u0439.<\/li>\n<li><strong>\u0414\u043e\u043a\u0430\u0437\u0430\u0442\u044c<\/strong> \u0441\u0432\u043e\u0439\u0441\u0442\u0432\u0430, \u043a\u043e\u0442\u043e\u0440\u044b\u0435 \u043f\u0440\u0438\u0432\u043e\u0434\u044f\u0442 \u043a \u0430\u043b\u0433\u0435\u0431\u0440\u0435 \u043f\u0440\u0435\u0434\u0435\u043b\u043e\u0432, \u0438\u0441\u043f\u043e\u043b\u044c\u0437\u0443\u044f \u0432\u044b\u0432\u043e\u0434\u044b <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\epsilon-\\delta<\/span><\/span>.<\/li>\n<li><strong>\u0412\u044b\u0447\u0438\u0441\u043b\u0438\u0442\u044c<\/strong> \u043f\u0440\u0435\u0434\u0435\u043b\u044b \u0444\u0443\u043d\u043a\u0446\u0438\u0439 \u043e\u0434\u043d\u043e\u0439 \u0434\u0435\u0439\u0441\u0442\u0432\u0438\u0442\u0435\u043b\u044c\u043d\u043e\u0439 \u043f\u0435\u0440\u0435\u043c\u0435\u043d\u043d\u043e\u0439, \u0438\u0441\u043f\u043e\u043b\u044c\u0437\u0443\u044f \u0430\u043b\u0433\u0435\u0431\u0440\u0443 \u043f\u0440\u0435\u0434\u0435\u043b\u043e\u0432 \u0438 \u0438\u0445 \u0441\u0432\u043e\u0439\u0441\u0442\u0432\u0430.<\/li>\n<\/ul>\n<p><\/em><\/p>\n<p><center><br \/>\n<strong>\u041e\u0413\u041b\u0410\u0412\u041b\u0415\u041d\u0418\u0415<\/strong><br \/>\n<a href=\"#1\"><strong>\u0412\u0432\u0435\u0434\u0435\u043d\u0438\u0435<\/strong><\/a><br \/>\n<a href=\"#2\"><strong>\u0418\u043d\u0442\u0443\u0438\u0442\u0438\u0432\u043d\u043e\u0435 \u043f\u0440\u0435\u0434\u0441\u0442\u0430\u0432\u043b\u0435\u043d\u0438\u0435 \u043e \u043f\u0440\u0435\u0434\u0435\u043b\u0435 \u0444\u0443\u043d\u043a\u0446\u0438\u0438 \u0441 \u0433\u0440\u0430\u0444\u0438\u0447\u0435\u0441\u043a\u043e\u0439 \u0442\u043e\u0447\u043a\u0438 \u0437\u0440\u0435\u043d\u0438\u044f<\/strong><\/a><br \/>\n<a href=\"#3\"><strong>\u0424\u043e\u0440\u043c\u0430\u043b\u044c\u043d\u043e\u0435 \u043e\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u0438\u0435 \u043f\u0440\u0435\u0434\u0435\u043b\u0430<\/strong><\/a><br \/>\n<a href=\"#4\"><strong>\u0421\u0432\u043e\u0439\u0441\u0442\u0432\u0430 \u043f\u0440\u0435\u0434\u0435\u043b\u043e\u0432<\/strong><\/a><br \/>\n<a href=\"#5\">\u0415\u0441\u043b\u0438 \u043f\u0440\u0435\u0434\u0435\u043b \u0441\u0443\u0449\u0435\u0441\u0442\u0432\u0443\u0435\u0442, \u0442\u043e \u043e\u043d \u0435\u0434\u0438\u043d\u0441\u0442\u0432\u0435\u043d\u043d\u044b\u0439<\/a><br \/>\n<a href=\"#6\">\u0410\u043b\u0433\u0435\u0431\u0440\u0430 \u043f\u0440\u0435\u0434\u0435\u043b\u043e\u0432<\/a><br \/>\n<a href=\"#7\">\u0412\u044b\u0447\u0438\u0441\u043b\u0435\u043d\u0438\u0435 \u043f\u0440\u043e\u0441\u0442\u044b\u0445 \u043f\u0440\u0435\u0434\u0435\u043b\u043e\u0432<\/a><br \/>\n<\/center><\/p>\n<p><center><iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/FEPfoAfPsFY\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/center><\/p>\n<p><a name=\"1\"><\/a><\/p>\n<h2>\u0412\u0432\u0435\u0434\u0435\u043d\u0438\u0435<\/h2>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=FEPfoAfPsFY&amp;t=240s\" rel=\"noopener\" target=\"_blank\"><strong><\u041a\u0430\u043a\u043e\u0432\u0430 \u0440\u0430\u0437\u043d\u0438\u0446\u0430 \u043c\u0435\u0436\u0434\u0443 \u0438\u0437\u0443\u0447\u0435\u043d\u0438\u0435\u043c \u0430\u043b\u0433\u0435\u0431\u0440\u044b \u0438 \u0433\u0435\u043e\u043c\u0435\u0442\u0440\u0438\u0438 \u043f\u043e \u0441\u0440\u0430\u0432\u043d\u0435\u043d\u0438\u044e \u0441 \u0438\u0437\u0443\u0447\u0435\u043d\u0438\u0435\u043c \u0430\u043d\u0430\u043b\u0438\u0437\u0430?<\/strong><\/a> \u041e\u0442\u0432\u0435\u0442 \u043d\u0430 \u044d\u0442\u043e\u0442 \u0432\u043e\u043f\u0440\u043e\u0441 \u0434\u0430\u0435\u0442 \u043a\u043e\u043d\u0446\u0435\u043f\u0446\u0438\u044f \u043f\u0440\u0435\u0434\u0435\u043b\u0430. \u0412 \u044d\u0442\u043e\u0439 \u0441\u0442\u0430\u0442\u044c\u0435 \u0440\u0430\u0441\u0441\u043c\u0430\u0442\u0440\u0438\u0432\u0430\u0435\u0442\u0441\u044f, \u0442\u0430\u043a\u0438\u043c \u043e\u0431\u0440\u0430\u0437\u043e\u043c, \u043f\u0440\u0435\u0434\u0435\u043b \u0438 \u0435\u0433\u043e \u043e\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u0438\u0435.<\/p>\n<p style=\"text-align: justify;\">\u0421\u043b\u043e\u0432\u043e \u00ab\u043f\u0440\u0435\u0434\u0435\u043b\u00bb \u043e\u0431\u044b\u0447\u043d\u043e \u0430\u0441\u0441\u043e\u0446\u0438\u0438\u0440\u0443\u0435\u0442\u0441\u044f \u0443 \u043d\u0430\u0441 \u0441 \u0447\u0435\u043c-\u0442\u043e \u0432\u0440\u043e\u0434\u0435 \u0433\u0440\u0430\u043d\u0438\u0446\u044b, \u043a\u0430\u043a \u0433\u0440\u0430\u043d\u0438\u0446\u0430 \u0438\u043d\u0442\u0435\u0440\u0432\u0430\u043b\u0430 \u0441 \u043a\u043e\u043d\u0446\u0430\u043c\u0438 a, b (\u043d\u0435\u0437\u0430\u0432\u0438\u0441\u0438\u043c\u043e \u043e\u0442 \u0438\u0445 \u043f\u0440\u0438\u0440\u043e\u0434\u044b)<\/p>\n<p style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">[a,b[\\;\\; ;\\;\\; ]a,b]\\;\\; ; \\;\\; ]a,b[\\;\\; ; [a,b] <\/span><\/span>,<\/p>\n<p style=\"text-align: justify;\">\u0438\u043b\u0438 \u043a\u0430\u043a \u0441 \u043d\u0430\u0441\u0442\u043e\u044f\u0449\u0438\u043c, \u043a\u043e\u0442\u043e\u0440\u044b\u0439 \u043c\u043e\u0436\u043d\u043e \u043d\u0430\u0437\u0432\u0430\u0442\u044c \u0433\u0440\u0430\u043d\u0438\u0446\u0435\u0439 \u043c\u0435\u0436\u0434\u0443 \u043f\u0440\u043e\u0448\u043b\u044b\u043c \u0438 \u0431\u0443\u0434\u0443\u0449\u0438\u043c. \u041f\u0440\u0438\u043c\u0435\u0440\u043d\u043e \u0442\u0430\u043a\u0438\u043c \u0436\u0435 \u043e\u0431\u0440\u0430\u0437\u043e\u043c \u043f\u043e\u043d\u044f\u0442\u0438\u0435 \u043f\u0440\u0435\u0434\u0435\u043b\u0430 \u0432\u0432\u043e\u0434\u0438\u0442 \u043c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u0447\u0435\u0441\u043a\u043e\u0435 \u043f\u043e\u043d\u0438\u043c\u0430\u043d\u0438\u0435 \u044d\u0442\u043e\u0439 \u0438\u043d\u0442\u0443\u0438\u0442\u0438\u0432\u043d\u043e\u0439 \u0438\u0434\u0435\u0438 \u0430\u0441\u0438\u043c\u043f\u0442\u043e\u0442\u0438\u0447\u0435\u0441\u043a\u043e\u0433\u043e \u043f\u0440\u0438\u0431\u043b\u0438\u0436\u0435\u043d\u0438\u044f \u043a \u043e\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u043d\u043e\u0439 \u0442\u043e\u0447\u043a\u0435.<\/p>\n<p><a name=\"2\"><\/a><\/p>\n<h2>\u0418\u043d\u0442\u0443\u0438\u0442\u0438\u0432\u043d\u043e\u0435 \u043f\u0440\u0435\u0434\u0441\u0442\u0430\u0432\u043b\u0435\u043d\u0438\u0435 \u043e \u043f\u0440\u0435\u0434\u0435\u043b\u0435 \u0444\u0443\u043d\u043a\u0446\u0438\u0438 \u0441 \u0433\u0440\u0430\u0444\u0438\u0447\u0435\u0441\u043a\u043e\u0439 \u0442\u043e\u0447\u043a\u0438 \u0437\u0440\u0435\u043d\u0438\u044f<\/h2>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=FEPfoAfPsFY&amp;t=314s\" rel=\"noopener\" target=\"_blank\"><strong>\u0427\u0442\u043e\u0431\u044b \u043d\u0430\u0447\u0430\u0442\u044c \u0432\u0438\u0437\u0443\u0430\u043b\u0438\u0437\u0438\u0440\u043e\u0432\u0430\u0442\u044c \u0438\u0434\u0435\u044e \u043f\u0440\u0435\u0434\u0435\u043b\u0430, \u0446\u0435\u043b\u0435\u0441\u043e\u043e\u0431\u0440\u0430\u0437\u043d\u043e \u043d\u0430\u0447\u0430\u0442\u044c \u0441 \u0433\u0440\u0430\u0444\u0438\u0447\u0435\u0441\u043a\u043e\u0433\u043e \u043f\u0440\u0435\u0434\u0441\u0442\u0430\u0432\u043b\u0435\u043d\u0438\u044f<\/strong><\/a> \u0444\u0443\u043d\u043a\u0446\u0438\u0438 \u0438 \u0437\u0430\u0434\u0430\u0442\u044c\u0441\u044f \u0432\u043e\u043f\u0440\u043e\u0441\u043e\u043c, \u0447\u0442\u043e \u043f\u0440\u043e\u0438\u0437\u043e\u0439\u0434\u0435\u0442 \u0441 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f(x)<\/span><\/span>, \u043a\u043e\u0433\u0434\u0430 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x<\/span><\/span> \u0431\u0443\u0434\u0435\u0442 \u043f\u0440\u0438\u0431\u043b\u0438\u0436\u0430\u0442\u044c\u0441\u044f \u043a <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x_0<\/span><\/span> \u043d\u0430\u0441\u0442\u043e\u043b\u044c\u043a\u043e \u0431\u043b\u0438\u0437\u043a\u043e, \u043d\u0430\u0441\u043a\u043e\u043b\u044c\u043a\u043e \u044d\u0442\u043e \u0432\u043e\u0437\u043c\u043e\u0436\u043d\u043e.<\/p>\n<p><center><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/1.bp.blogspot.com\/-rMjBnCIK8Ts\/YGDfXYswS4I\/AAAAAAAAEwA\/1GY0wy3JkXk99kveDTp1SltJOTAITgN3wCLcBGAsYHQ\/s0\/limite.PNG\" alt=\"\u043f\u0440\u0435\u0434\u0435\u043b \u0444\u0443\u043d\u043a\u0446\u0438\u0438\" class=\"alignnone size-full lazyload\" width=\"692\" height=\"565\" \/><noscript><img decoding=\"async\" src=\"https:\/\/1.bp.blogspot.com\/-rMjBnCIK8Ts\/YGDfXYswS4I\/AAAAAAAAEwA\/1GY0wy3JkXk99kveDTp1SltJOTAITgN3wCLcBGAsYHQ\/s0\/limite.PNG\" alt=\"\u043f\u0440\u0435\u0434\u0435\u043b \u0444\u0443\u043d\u043a\u0446\u0438\u0438\" class=\"alignnone size-full lazyload\" width=\"692\" height=\"565\" \/><\/noscript><\/center><\/p>\n<p style=\"text-align: justify;\">\u0415\u0441\u043b\u0438 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x<\/span><\/span> \u043d\u0430\u0445\u043e\u0434\u0438\u0442\u0441\u044f \u0440\u044f\u0434\u043e\u043c \u0441 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x_0<\/span><\/span>, \u0442\u043e \u0441\u0443\u0449\u0435\u0441\u0442\u0432\u0443\u0435\u0442 \u043e\u0442\u043a\u0440\u044b\u0442\u044b\u0439 \u0438\u043d\u0442\u0435\u0440\u0432\u0430\u043b \u0440\u0430\u0434\u0438\u0443\u0441\u0430 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\delta<\/span><\/span> \u0438 \u0441 \u0446\u0435\u043d\u0442\u0440\u043e\u043c \u0432 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x_0<\/span><\/span>, \u0432 \u043a\u043e\u0442\u043e\u0440\u043e\u043c \u0441\u043e\u0434\u0435\u0440\u0436\u0438\u0442\u0441\u044f <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x<\/span><\/span>. \u042d\u0442\u043e \u043c\u043e\u0436\u043d\u043e \u043f\u0440\u0435\u0434\u0441\u0442\u0430\u0432\u0438\u0442\u044c \u0442\u0440\u0435\u043c\u044f \u0440\u0430\u0437\u043d\u044b\u043c\u0438 \u0441\u043f\u043e\u0441\u043e\u0431\u0430\u043c\u0438:<\/p>\n<p style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">|x-x_0|\\lt \\delta<\/span><\/span>,<\/p>\n<p style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">|x\\in]x_0 - \\delta , x_0 + \\delta[ <\/span><\/span>,<\/p>\n<p style=\"text-align: center;\">\u0438\u043b\u0438 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x\\in\\mathcal{B}(x_0,\\delta)<\/span><\/span><\/p>\n<p style=\"text-align: justify;\"><em>\u0412 \u043d\u0430\u0448\u0435\u043c \u043a\u043e\u043d\u0442\u0435\u043a\u0441\u0442\u0435 \u044d\u0442\u043e \u0442\u0440\u0438 \u0441\u043f\u043e\u0441\u043e\u0431\u0430 \u0441\u043a\u0430\u0437\u0430\u0442\u044c \u043e\u0434\u043d\u043e \u0438 \u0442\u043e \u0436\u0435; \u0445\u043e\u0442\u044f \u043f\u043e\u0441\u043b\u0435\u0434\u043d\u0438\u0439, \u043a\u043e\u0442\u043e\u0440\u044b\u0439 \u0447\u0438\u0442\u0430\u0435\u0442\u0441\u044f \u043a\u0430\u043a \u00abx \u0441\u043e\u0434\u0435\u0440\u0436\u0438\u0442\u0441\u044f \u0432 \u043e\u0442\u043a\u0440\u044b\u0442\u043e\u043c \u0448\u0430\u0440\u0435 \u0441 \u0446\u0435\u043d\u0442\u0440\u043e\u043c \u0432 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x_0<\/span><\/span> \u0438 \u0440\u0430\u0434\u0438\u0443\u0441\u043e\u043c <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\delta<\/span><\/span>, \u0431\u044b\u043b \u0431\u044b \u0431\u043e\u043b\u0435\u0435 \u043f\u043e\u0434\u0445\u043e\u0434\u044f\u0449\u0438\u043c \u0434\u043b\u044f <strong>\u043a\u0443\u0440\u0441\u0430 \u0442\u043e\u043f\u043e\u043b\u043e\u0433\u0438\u0438<\/strong>, \u0433\u0434\u0435 \u044d\u0442\u0430 \u0442\u0435\u043c\u0430 \u00ab\u0431\u043b\u0438\u0437\u043e\u0441\u0442\u0438\u00bb \u0431\u0443\u0434\u0435\u0442 \u0440\u0430\u0441\u0441\u043c\u0430\u0442\u0440\u0438\u0432\u0430\u0442\u044c\u0441\u044f \u0431\u043e\u043b\u0435\u0435 \u0433\u043b\u0443\u0431\u043e\u043a\u043e.<\/em><\/p>\n<p style=\"text-align: justify;\">\u0415\u0441\u043b\u0438 \u044d\u0442\u043e \u0443\u0441\u043b\u043e\u0432\u0438\u0435 \u0432\u044b\u043f\u043e\u043b\u043d\u044f\u0435\u0442\u0441\u044f, \u0442\u043e \u043c\u044b \u0443\u0432\u0438\u0434\u0438\u043c, \u0447\u0442\u043e \u0441\u0443\u0449\u0435\u0441\u0442\u0432\u0443\u0435\u0442 \u0434\u0440\u0443\u0433\u043e\u0439 \u043e\u0442\u043a\u0440\u044b\u0442\u044b\u0439 \u0438\u043d\u0442\u0435\u0440\u0432\u0430\u043b \u0441 \u0446\u0435\u043d\u0442\u0440\u043e\u043c \u0432 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">l<\/span><\/span> \u0438 \u0440\u0430\u0434\u0438\u0443\u0441\u043e\u043c <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\epsilon<\/span><\/span>, \u0432 \u043a\u043e\u0442\u043e\u0440\u043e\u043c \u0441\u043e\u0434\u0435\u0440\u0436\u0438\u0442\u0441\u044f <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f(x)<\/span><\/span>, \u0442\u043e \u0435\u0441\u0442\u044c: <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">|f(x) - l|\\lt \\epsilon<\/span><\/span>.<\/p>\n<p><center><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/1.bp.blogspot.com\/-26xU-o1y-Eg\/YGDfXYgOp2I\/AAAAAAAAEwE\/FGMGEQdvRzg_OvnUqKolJ9v51xUVF4O7QCLcBGAsYHQ\/s0\/limite2.PNG\" alt=\"\u043f\u0440\u0435\u0434\u0435\u043b \u0444\u0443\u043d\u043a\u0446\u0438\u0438\" class=\"alignnone size-full lazyload\" width=\"625\" height=\"549\" \/><noscript><img decoding=\"async\" src=\"https:\/\/1.bp.blogspot.com\/-26xU-o1y-Eg\/YGDfXYgOp2I\/AAAAAAAAEwE\/FGMGEQdvRzg_OvnUqKolJ9v51xUVF4O7QCLcBGAsYHQ\/s0\/limite2.PNG\" alt=\"\u043f\u0440\u0435\u0434\u0435\u043b \u0444\u0443\u043d\u043a\u0446\u0438\u0438\" class=\"alignnone size-full lazyload\" width=\"625\" height=\"549\" \/><\/noscript><\/center><\/p>\n<p style=\"text-align: justify;\">\u041e\u0442\u0441\u044e\u0434\u0430 \u0432\u043e\u0437\u043d\u0438\u043a\u0430\u0435\u0442 \u043e\u0441\u043d\u043e\u0432\u043d\u0430\u044f \u0438\u0434\u0435\u044f \u043c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u0447\u0435\u0441\u043a\u043e\u0433\u043e \u043f\u043e\u043d\u044f\u0442\u0438\u044f \u043f\u0440\u0435\u0434\u0435\u043b\u0430, \u0437\u0430\u043a\u043b\u044e\u0447\u0430\u044e\u0449\u0430\u044f\u0441\u044f \u0432 \u0442\u043e\u043c, \u0447\u0442\u043e \u043f\u0440\u0435\u0434\u0435\u043b \u0441\u0443\u0449\u0435\u0441\u0442\u0432\u0443\u0435\u0442, \u0435\u0441\u043b\u0438: \u043f\u0440\u0438 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0 \\lt|x-x_0|\\lt \\delta<\/span><\/span> \u0432\u044b\u043f\u043e\u043b\u043d\u044f\u0435\u0442\u0441\u044f <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">|f(x)-l|\\lt \\epsilon<\/span><\/span>; \u0438 \u044d\u0442\u043e \u0437\u043d\u0430\u0447\u0435\u043d\u0438\u0435 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">l<\/span><\/span> \u0431\u0443\u0434\u0435\u0442 \u043f\u0440\u0435\u0434\u0435\u043b\u043e\u043c \u0444\u0443\u043d\u043a\u0446\u0438\u0438, \u043a\u043e\u0433\u0434\u0430 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x<\/span><\/span> \u0441\u0442\u0440\u0435\u043c\u0438\u0442\u0441\u044f \u043a <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x_0<\/span><\/span> \u043d\u0430\u0441\u0442\u043e\u043b\u044c\u043a\u043e \u0431\u043b\u0438\u0437\u043a\u043e, \u043d\u0430\u0441\u043a\u043e\u043b\u044c\u043a\u043e \u043c\u044b \u0437\u0430\u0445\u043e\u0442\u0438\u043c.<\/p>\n<p><a name=\"3\"><\/a><\/p>\n<h2>\u0424\u043e\u0440\u043c\u0430\u043b\u044c\u043d\u043e\u0435 \u043e\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u0438\u0435 \u043f\u0440\u0435\u0434\u0435\u043b\u0430<\/h2>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=FEPfoAfPsFY&amp;t=689s\" rel=\"noopener\" target=\"_blank\"><strong>\u041e\u0442 \u0438\u043d\u0442\u0443\u0438\u0442\u0438\u0432\u043d\u043e\u0433\u043e \u0438 \u0433\u0440\u0430\u0444\u0438\u0447\u0435\u0441\u043a\u043e\u0433\u043e \u043f\u0440\u0435\u0434\u0441\u0442\u0430\u0432\u043b\u0435\u043d\u0438\u044f, \u043a\u043e\u0442\u043e\u0440\u043e\u0435 \u0431\u044b\u043b\u043e \u0442\u043e\u043b\u044c\u043a\u043e \u0447\u0442\u043e \u043f\u0440\u0435\u0434\u0441\u0442\u0430\u0432\u043b\u0435\u043d\u043e, \u043c\u043e\u0436\u043d\u043e \u043f\u0435\u0440\u0435\u0439\u0442\u0438 \u043a \u0444\u043e\u0440\u043c\u0430\u043b\u044c\u043d\u043e\u043c\u0443 \u043e\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u0438\u044e \u043f\u0440\u0435\u0434\u0435\u043b\u0430.<\/strong> <\/a>\u041c\u044b \u0433\u043e\u0432\u043e\u0440\u0438\u043c, \u0447\u0442\u043e \u043f\u0440\u0435\u0434\u0435\u043b \u0441\u0443\u0449\u0435\u0441\u0442\u0432\u0443\u0435\u0442, \u043a\u043e\u0433\u0434\u0430, \u043d\u0435\u0437\u0430\u0432\u0438\u0441\u0438\u043c\u043e \u043e\u0442 \u0442\u043e\u0433\u043e, \u043a\u0430\u043a\u043e\u0435 \u0437\u043d\u0430\u0447\u0435\u043d\u0438\u0435 \u043f\u0440\u0438\u043c\u0435\u0442 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\epsilon<\/span><\/span> (\u0442\u043e \u0435\u0441\u0442\u044c \u0440\u0430\u0441\u0441\u0442\u043e\u044f\u043d\u0438\u0435 \u043c\u0435\u0436\u0434\u0443 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f(x)<\/span><\/span> \u0438 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">l<\/span><\/span>), \u0432\u0441\u0435\u0433\u0434\u0430 \u0441\u0443\u0449\u0435\u0441\u0442\u0432\u0443\u0435\u0442 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\delta<\/span><\/span> \u0442\u0430\u043a\u043e\u0435, \u0447\u0442\u043e \u0435\u0441\u043b\u0438 \u0432\u044b\u043f\u043e\u043b\u043d\u044f\u0435\u0442\u0441\u044f <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0 \\lt|x-x_0|\\lt \\delta<\/span><\/span>, \u0442\u043e \u0432\u044b\u043f\u043e\u043b\u043d\u044f\u0435\u0442\u0441\u044f \u0438 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">|f(x) - l|\\lt \\epsilon.<\/span><\/span> \u042d\u0442\u0430 \u0438\u0434\u0435\u044f, \u043a\u043e\u0442\u043e\u0440\u0430\u044f \u0438\u0437\u043d\u0430\u0447\u0430\u043b\u044c\u043d\u043e \u043c\u043e\u0436\u0435\u0442 \u0431\u044b\u0442\u044c \u0442\u0440\u0443\u0434\u043d\u043e\u0439 \u0434\u043b\u044f \u043f\u043e\u043d\u0438\u043c\u0430\u043d\u0438\u044f \u0438 \u043f\u0440\u0438\u0447\u0438\u043d\u043e\u0439 \u0441\u043b\u0435\u0437 \u0434\u043b\u044f \u0431\u043e\u043b\u044c\u0448\u0438\u043d\u0441\u0442\u0432\u0430 \u0441\u0442\u0443\u0434\u0435\u043d\u0442\u043e\u0432 \u043f\u043e \u0432\u0441\u0435\u043c\u0443 \u043c\u0438\u0440\u0443, \u0438\u0437\u0443\u0447\u0430\u044e\u0449\u0438\u0445 \u043c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u0447\u0435\u0441\u043a\u0438\u0439 \u0430\u043d\u0430\u043b\u0438\u0437, \u043c\u043e\u0436\u0435\u0442 \u0431\u044b\u0442\u044c \u043f\u0440\u0435\u0434\u0441\u0442\u0430\u0432\u043b\u0435\u043d\u0430 \u0441\u043b\u0435\u0434\u0443\u044e\u0449\u0435\u0439 \u0444\u043e\u0440\u043c\u0443\u043b\u043e\u0439:<\/p>\n<p style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to x_0}f(x)=l := \\left(\\forall \\epsilon \\gt 0\\right)\\left(\\exists \\delta\\gt 0\\right) \\left(0 \\lt|x-x_0|\\lt\\delta \\rightarrow |f(x) - l|\\lt \\epsilon\\right)<\/span><\/span>,<\/p>\n<p><a name=\"4\"><\/a><\/p>\n<h2>\u0421\u0432\u043e\u0439\u0441\u0442\u0432\u0430 \u043f\u0440\u0435\u0434\u0435\u043b\u043e\u0432<\/h2>\n<p style=\"text-align: justify;\">\u0412\u0430\u0436\u043d\u043e\u0441\u0442\u044c \u0444\u043e\u0440\u043c\u0430\u043b\u044c\u043d\u043e\u0433\u043e \u043e\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u0438\u044f \u043f\u0440\u0435\u0434\u0435\u043b\u0430 \u0437\u0430\u043a\u043b\u044e\u0447\u0430\u0435\u0442\u0441\u044f \u0432 \u0442\u043e\u043c, \u0447\u0442\u043e \u0442\u0435\u043f\u0435\u0440\u044c, \u0438\u0441\u0445\u043e\u0434\u044f \u0438\u0437 \u044d\u0442\u043e\u0433\u043e, \u043c\u044b \u043c\u043e\u0436\u0435\u043c \u0434\u043e\u043a\u0430\u0437\u0430\u0442\u044c \u0435\u0433\u043e \u0441\u0432\u043e\u0439\u0441\u0442\u0432\u0430, \u043a\u0430\u043a \u0442\u0435, \u043a\u043e\u0442\u043e\u0440\u044b\u0435 \u043a\u0430\u0436\u0443\u0442\u0441\u044f \u0438\u043d\u0442\u0443\u0438\u0442\u0438\u0432\u043d\u043e \u043f\u043e\u043d\u044f\u0442\u043d\u044b\u043c\u0438, \u0442\u0430\u043a \u0438 \u0442\u0435, \u043a\u043e\u0442\u043e\u0440\u044b\u0435 \u043c\u043e\u0433\u0443\u0442 \u043d\u0435 \u0431\u044b\u0442\u044c \u0442\u0430\u043a\u0438\u043c\u0438 \u043e\u0447\u0435\u0432\u0438\u0434\u043d\u044b\u043c\u0438.<\/p>\n<p style=\"text-align: justify;\">\u041f\u0440\u0435\u0436\u0434\u0435 \u0447\u0435\u043c \u043f\u0440\u043e\u0434\u043e\u043b\u0436\u0438\u0442\u044c, \u0445\u043e\u0442\u044f \u044d\u0442\u043e \u043d\u0435 \u0441\u0442\u0440\u043e\u0433\u043e \u043d\u0435\u043e\u0431\u0445\u043e\u0434\u0438\u043c\u043e, \u043d\u0430\u0441\u0442\u043e\u044f\u0442\u0435\u043b\u044c\u043d\u043e \u0440\u0435\u043a\u043e\u043c\u0435\u043d\u0434\u0443\u0435\u0442\u0441\u044f \u0438\u0437\u0443\u0447\u0438\u0442\u044c \u043d\u0435\u043a\u043e\u0442\u043e\u0440\u044b\u0435 \u043a\u043e\u043d\u0446\u0435\u043f\u0446\u0438\u0438 <a href=\"http:\/\/toposuranos.com\/material\/es\/category\/matematica\/logica-matematica\/logica-proposicional\/\" rel=\"noopener\" target=\"_blank\"><strong>\u043c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u0447\u0435\u0441\u043a\u043e\u0439 \u043b\u043e\u0433\u0438\u043a\u0438<\/strong><\/a>, \u0447\u0442\u043e\u0431\u044b \u043b\u0443\u0447\u0448\u0435 \u043f\u043e\u043d\u044f\u0442\u044c \u0434\u043e\u043a\u0430\u0437\u0430\u0442\u0435\u043b\u044c\u0441\u0442\u0432\u0430, \u043f\u0440\u0438\u0432\u0435\u0434\u0435\u043d\u043d\u044b\u0435 \u043d\u0438\u0436\u0435.<\/p>\n<p><a name=\"5\"><\/a><\/p>\n<h3>\u0415\u0441\u043b\u0438 \u043f\u0440\u0435\u0434\u0435\u043b \u0441\u0443\u0449\u0435\u0441\u0442\u0432\u0443\u0435\u0442, \u0442\u043e \u043e\u043d \u0435\u0434\u0438\u043d\u0441\u0442\u0432\u0435\u043d\u043d\u044b\u0439<\/h3>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=FEPfoAfPsFY&amp;t=904s\" rel=\"noopener\" target=\"_blank\"><strong>\u0427\u0442\u043e\u0431\u044b \u0434\u043e\u043a\u0430\u0437\u0430\u0442\u044c \u044d\u0442\u043e \u0441\u0432\u043e\u0439\u0441\u0442\u0432\u043e, \u043c\u044b \u0432\u043e\u0441\u043f\u043e\u043b\u044c\u0437\u0443\u0435\u043c\u0441\u044f \u043c\u0435\u0442\u043e\u0434\u043e\u043c \u0434\u043e\u043a\u0430\u0437\u0430\u0442\u0435\u043b\u044c\u0441\u0442\u0432\u0430 \u043e\u0442 \u043f\u0440\u043e\u0442\u0438\u0432\u043d\u043e\u0433\u043e.<\/strong><\/a> \u041d\u0430\u0447\u043d\u0435\u043c \u0441 \u043e\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u0438\u044f \u0441\u043b\u0435\u0434\u0443\u044e\u0449\u0435\u0433\u043e \u043c\u043d\u043e\u0436\u0435\u0441\u0442\u0432\u0430 \u043f\u0440\u0435\u0434\u043f\u043e\u0441\u044b\u043b\u043e\u043a:<\/p>\n<p style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle\\mathcal{H}= \\{\\lim_{x\\to x_0}f(x) = L, \\lim_{x\\to x_0}f(x) = L^\\prime, L\\neq L^\\prime\\}<\/span><\/span>.<\/p>\n<p style=\"text-align: justify;\">\u0418\u0441\u0445\u043e\u0434\u044f \u0438\u0437 \u044d\u0442\u043e\u0433\u043e, \u043c\u043e\u0436\u043d\u043e \u043f\u043e\u0441\u0442\u0440\u043e\u0438\u0442\u044c \u0441\u043b\u0435\u0434\u0443\u044e\u0449\u0435\u0435 \u0444\u043e\u0440\u043c\u0430\u043b\u044c\u043d\u043e\u0435 \u0434\u043e\u043a\u0430\u0437\u0430\u0442\u0435\u043b\u044c\u0441\u0442\u0432\u043e:<\/p>\n<table>\n<tbody>\n<tr>\n<td style=\" text-align: right;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(1)<\/span><\/span><\/td>\n<td style=\" text-align: left;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\mathcal{H}\\vdash \\lim_{x\\to x_0}f(x) = L <\/span><\/span>; <strong>\u041f\u0440\u0435\u0434\u043f\u043e\u043b\u043e\u0436\u0435\u043d\u0438\u0435<\/strong><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td style=\" text-align: left;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\mathcal{H}\\vdash \\left(\\forall \\epsilon \\gt 0\\right)\\left(\\exists \\delta\\gt 0\\right) \\left(0 \\lt|x-x_0|\\lt\\delta \\rightarrow |f(x) - L|\\lt \\epsilon\\right) <\/span><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\" text-align: right;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(2)<\/span><\/span><\/td>\n<td style=\" text-align: left;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\mathcal{H}\\vdash \\lim_{x\\to x_0}f(x) = L^\\prime <\/span><\/span>; \u041f\u0440\u0435\u0434\u043f\u043e\u043b\u043e\u0436\u0435\u043d\u0438\u0435<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td style=\" text-align: left;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\mathcal{H}\\vdash \\left(\\forall \\epsilon \\gt 0\\right)\\left(\\exists \\delta\\gt 0\\right) \\left(0 \\lt|x-x_0|\\lt\\delta \\rightarrow |f(x) - L^\\prime |\\lt \\epsilon\\right) <\/span><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\" text-align: right;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(3)<\/span><\/span><\/td>\n<td style=\" text-align: left;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\mathcal{H}\\vdash L \\neq L^\\prime <\/span><\/span>; \u041f\u0440\u0435\u0434\u043f\u043e\u043b\u043e\u0436\u0435\u043d\u0438\u0435<\/td>\n<\/tr>\n<tr>\n<td style=\" text-align: right;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(4)<\/span><\/span><\/td>\n<td style=\" text-align: left;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\mathcal{H}\\vdash \\left(\\forall \\epsilon \\gt 0\\right)\\left(\\exists \\delta\\gt 0\\right) \\left(0 \\lt|x-x_0|\\lt\\delta \\rightarrow\\right.<\/span><\/span> <span style=\"background-color: #ffff80; color:#000000\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\left. \\left[ \\left( |f(x) - L |\\lt \\epsilon \\right) \\wedge \\left( |f(x) - L^\\prime |\\lt \\epsilon\\right) \\right] \\right. <\/span><\/span><\/span><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">)<\/span><\/span>; <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\wedge<\/span><\/span>&#8211;<strong>\u0418\u043d\u0442.<\/strong>(1,2)<\/td>\n<\/tr>\n<tr>\n<td style=\" text-align: right;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(5)<\/span><\/span><\/td>\n<td style=\" text-align: left;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\mathcal{H}\\cup\\{L\\lt L^\\prime\\}\\vdash \\left(\\forall \\epsilon \\gt 0\\right)\\left(\\exists \\delta\\gt 0\\right) \\left(0 \\lt|x-x_0|\\lt\\delta \\rightarrow\\right.<\/span><\/span> <span style=\"background-color: #ffff80; color:#000000\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\left. \\left[ \\left( |f(x) - L |\\lt \\epsilon \\right) \\wedge \\left( |f(x) - L^\\prime |\\lt \\epsilon\\right) \\right] \\right. <\/span><\/span><\/span><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">)<\/span><\/span>; <strong>\u041c\u043e\u043d\u043e\u0442\u043e\u043d\u0438\u044f<\/strong>(4)<\/td>\n<\/tr>\n<tr>\n<td style=\" text-align: right;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(6)<\/span><\/span><\/td>\n<td style=\" text-align: left;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\mathcal{H}\\cup\\{L\\lt L^\\prime\\}\\vdash \\epsilon = \\frac{L - L^\\prime}{2}\\gt 0 <\/span><\/span>; \u041f\u043e\u0441\u043a\u043e\u043b\u044c\u043a\u0443 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">L \\lt L^\\prime <\/span><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\" text-align: right;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(7)<\/span><\/span><\/td>\n<td style=\" text-align: left;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\mathcal{H}\\cup\\{L\\lt L^\\prime\\}\\vdash \\left(\\exists \\delta\\gt 0\\right) \\left(0 \\lt|x-x_0|\\lt\\delta \\rightarrow\\right.<\/span><\/span> <span style=\"background-color: #ffff80; color:#000000\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\left. \\left[ \\left( |f(x) - L |\\lt \\frac{L - L^\\prime}{2} \\right) \\wedge \\left( |f(x) - L^\\prime |\\lt \\frac{L - L^\\prime}{2}\\right) \\right] \\right. <\/span><\/span><\/span><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">)<\/span><\/span>; \u0418\u0441\u043f\u043e\u043b\u044c\u0437\u0443\u044f(5,6)<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td style=\" text-align: left;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\mathcal{H}\\cup\\{L\\lt L^\\prime\\}\\vdash (\\exists \\delta\\gt 0) (0 \\lt|x-x_0|\\lt\\delta \\rightarrow [<\/span><\/span> <span style=\"background-color: #a0ffa0; color:#000000\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">( 2 |f(x) - L |\\lt L - L^\\prime )<\/span><\/span><\/span> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\wedge<\/span><\/span> <span style=\"background-color: #a0a0ff; color:#000000\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">( 2|f(x) - L^\\prime |\\lt L - L^\\prime)<\/span><\/span><\/span> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\"> ])<\/span><\/span><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td style=\" text-align: left;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\mathcal{H}\\cup\\{L\\lt L^\\prime\\}\\vdash (\\exists \\delta\\gt 0) (0 \\lt|x-x_0|\\lt\\delta \\rightarrow [<\/span><\/span> <span style=\"background-color: #a0ffa0; color:#000000\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">( -L + L^\\prime \\lt 2 (f(x) - L )\\lt L - L^\\prime )<\/span><\/span><\/span> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\wedge<\/span><\/span> <span style=\"background-color: #a0a0ff; color:#000000\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">( -L + L^\\prime \\lt 2(f(x) - L^\\prime )\\lt L - L^\\prime)<\/span><\/span><\/span> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\"> ])<\/span><\/span><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td style=\" text-align: left;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\mathcal{H}\\cup\\{L\\lt L^\\prime\\}\\vdash (\\exists \\delta\\gt 0) (0 \\lt|x-x_0|\\lt\\delta \\rightarrow [<\/span><\/span> <span style=\"background-color: #a0ffa0; color:#000000\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">( -L + L^\\prime \\lt 2f(x) - 2L \\lt L - L^\\prime )<\/span><\/span><\/span> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\wedge<\/span><\/span> <span style=\"background-color: #a0a0ff; color:#000000\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">( -L + L^\\prime \\lt 2f(x) - 2L^\\prime \\lt L - \u041b^\\prime)<\/span><\/span><\/span> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\"> ])<\/span><\/span><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td style=\" text-align: left;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\mathcal{H}\\cup\\{L\\lt L^\\prime\\}\\vdash (\\exists \\delta\\gt 0) (0 \\lt|x-x_0|\\lt\\delta \\rightarrow [<\/span><\/span> <span style=\"background-color: #a0ffa0; color:#000000\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">( L + L^\\prime \\lt 2f(x) \\lt 3L - L^\\prime )<\/span><\/span><\/span> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\wedge<\/span><\/span> <span style=\"background-color: #a0a0ff; color:#000000\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">( -L + 3L^\\prime \\lt 2f(x) \\lt L + L^\\prime)<\/span><\/span><\/span> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\"> ])<\/span><\/span><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td style=\" text-align: left;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\mathcal{H}\\cup\\{L\\lt L^\\prime\\}\\vdash (\\exists \\delta\\gt 0) (0 \\lt|x-x_0|\\lt\\delta \\rightarrow [<\/span><\/span> <span style=\"background-color: #a0a0ff; color:#000000\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">( -L + 3L^\\prime \\lt 2f(x) \\lt L + L^\\prime)<\/span><\/span><\/span> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\wedge<\/span><\/span> <span style=\"background-color: #a0ffa0; color:#000000\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">( L + L^\\prime \\lt 2f(x) \\lt 3L - L^\\prime )<\/span><\/span><\/span> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\"> ])<\/span><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\" text-align: right;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(8)<\/span><\/span><\/td>\n<td style=\" text-align: left;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\mathcal{H}\\cup\\{L\\lt L^\\prime\\}\\vdash \\bot <\/span><\/span>; \u0418\u0437(1,2,6,7)<\/td>\n<\/tr>\n<tr>\n<td style=\" text-align: right;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(9)<\/span><\/span><\/td>\n<td style=\" text-align: left;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\mathcal{H}\\cup\\{L\\gt L^\\prime\\}\\vdash \\bot <\/span><\/span>; \u0410\u043d\u0430\u043b\u043e\u0433\u0438\u0447\u043d\u043e (8)<\/td>\n<\/tr>\n<tr>\n<td style=\" text-align: right;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(10)<\/span><\/span><\/td>\n<td style=\" text-align: left;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\mathcal{H}\\vdash [(L\\lt L^\\prime) \\vee (L\\gt L^\\prime)] \\rightarrow \\bot <\/span><\/span>; <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vee<\/span><\/span>-\u0438\u043d\u0442.(8,9)<\/td>\n<\/tr>\n<tr>\n<td style=\" text-align: right;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(11)<\/span><\/span><\/td>\n<td style=\" text-align: left;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\mathcal{H}\\vdash [L\\ \\neq L^\\prime] \\rightarrow \\bot <\/span><\/span>; \u041e\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u0438\u0435(10)<\/td>\n<\/tr>\n<tr>\n<td style=\" text-align: right;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(12)<\/span><\/span><\/td>\n<td style=\" text-align: left;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\mathcal{H}\\vdash \\bot <\/span><\/span>; <strong>MP<\/strong>(3,11)<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td style=\" text-align: left;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\left\\{\\lim_{x\\to x_0}f(x) = L, \\lim_{x\\to x_0}f(x) = L^\\prime, L\\neq L^\\prime\\right\\} \\vdash \\bot <\/span><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\" text-align: right;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(13)<\/span><\/span><\/td>\n<td style=\" text-align: left;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\left\\{\\lim_{x\\to x_0}f(x) = L, \\lim_{x\\to x_0}f(x) = L^\\prime \\right\\} \\vdash \\neg(L\\neq L^\\prime) <\/span><\/span>; <strong>\u041f\u0440\u043e\u0442\u0438\u0432\u043e\u0440\u0435\u0447\u0438\u0435<\/strong>(12)<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td style=\" text-align: left;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\left\\{\\lim_{x\\to x_0}f(x) = L, \\lim_{x\\to x_0}f(x) = L^\\prime \\right\\} \\vdash L = L^\\prime.<\/span><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: justify;\">\u0418\u0437 \u044d\u0442\u043e\u0433\u043e \u0434\u043e\u043a\u0430\u0437\u0430\u0442\u0435\u043b\u044c\u0441\u0442\u0432\u0430 \u043c\u044b \u043f\u043e\u043b\u0443\u0447\u0430\u0435\u043c, \u0447\u0442\u043e \u0435\u0441\u043b\u0438 \u0441\u0443\u0449\u0435\u0441\u0442\u0432\u0443\u0435\u0442 \u0434\u0432\u0430 \u043f\u0440\u0435\u0434\u0435\u043b\u0430, \u0442\u043e \u043e\u043d\u0438 \u0440\u0430\u0432\u043d\u044b, \u0438, \u0441\u043b\u0435\u0434\u043e\u0432\u0430\u0442\u0435\u043b\u044c\u043d\u043e, \u043f\u0440\u0435\u0434\u0435\u043b \u0435\u0434\u0438\u043d\u0441\u0442\u0432\u0435\u043d\u0435\u043d.<\/p>\n<p><a name=\"6\"><\/a><\/p>\n<h3>\u0410\u043b\u0433\u0435\u0431\u0440\u0430 \u041f\u0440\u0435\u0434\u0435\u043b\u043e\u0432<\/h3>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=FEPfoAfPsFY&amp;t=2011s\" rel=\"noopener\" target=\"_blank\"><strong>\u0421 \u0442\u0435\u043c, \u0447\u0442\u043e \u043c\u044b \u0438\u0437\u0443\u0447\u0438\u043b\u0438 \u0434\u043e \u0441\u0438\u0445 \u043f\u043e\u0440, \u043c\u044b \u0440\u0430\u0441\u0441\u043c\u043e\u0442\u0440\u0435\u043b\u0438 \u043e\u0441\u043d\u043e\u0432\u043d\u0443\u044e \u043c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u0447\u0435\u0441\u043a\u0443\u044e \u0438\u0434\u0435\u044e \u043f\u0440\u0435\u0434\u0435\u043b\u0430.<\/strong><\/a> \u041d\u043e \u0442\u043e\u043b\u044c\u043a\u043e \u044d\u0442\u043e\u0433\u043e \u043d\u0435\u0434\u043e\u0441\u0442\u0430\u0442\u043e\u0447\u043d\u043e, \u0447\u0442\u043e\u0431\u044b \u0432\u044b\u043f\u043e\u043b\u043d\u044f\u0442\u044c \u0440\u0430\u0441\u0447\u0435\u0442\u044b \u0441 \u043f\u0440\u0435\u0434\u0435\u043b\u0430\u043c\u0438 \u2014 \u0434\u0430\u0436\u0435 \u0431\u043b\u0438\u0437\u043a\u043e. \u0422\u043e\u043b\u044c\u043a\u043e \u0441\u0443\u043c\u0430\u0441\u0448\u0435\u0434\u0448\u0438\u0439, \u0436\u0430\u0436\u0434\u0443\u0449\u0438\u0439 \u0441\u0442\u0440\u0430\u0434\u0430\u043d\u0438\u0439, \u0431\u0443\u0434\u0435\u0442 \u0438\u0441\u043f\u043e\u043b\u044c\u0437\u043e\u0432\u0430\u0442\u044c \u043e\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u0438\u0435 \u043f\u0440\u0435\u0434\u0435\u043b\u0430 \u0434\u043b\u044f \u044d\u0442\u043e\u0433\u043e. \u0427\u0442\u043e\u0431\u044b \u0440\u0435\u0448\u0438\u0442\u044c \u044d\u0442\u0443 \u043f\u0440\u043e\u0431\u043b\u0435\u043c\u0443, \u0442\u0435\u043f\u0435\u0440\u044c \u043c\u044b \u0440\u0430\u0441\u0441\u043c\u043e\u0442\u0440\u0438\u043c \u0442\u0435\u0445\u043d\u0438\u043a\u0438, \u043a\u043e\u0442\u043e\u0440\u044b\u0435 \u043f\u043e\u043c\u043e\u0433\u0443\u0442 \u043d\u0430\u043c \u043d\u0430\u0447\u0430\u0442\u044c \u0432\u044b\u0447\u0438\u0441\u043b\u044f\u0442\u044c \u043d\u0435\u043a\u043e\u0442\u043e\u0440\u044b\u0435 \u043f\u0440\u0435\u0434\u0435\u043b\u044b.<\/p>\n<p style=\"text-align: justify;\">\u041f\u0443\u0441\u0442\u044c <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x_0, \\alpha, \\beta, L, M \\in \\mathbb{R},<\/span><\/span> \u0438 \u043f\u0443\u0441\u0442\u044c f \u0438 g \u2014 \u044d\u0442\u043e \u0434\u0435\u0439\u0441\u0442\u0432\u0438\u0442\u0435\u043b\u044c\u043d\u044b\u0435 \u0444\u0443\u043d\u043a\u0446\u0438\u0438, \u0442\u0430\u043a\u0438\u0435 \u0447\u0442\u043e:<\/p>\n<p style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to x_0} f(x) = L<\/span><\/span><\/p>\n<p style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to x_0} g(x) = M<\/span><\/span><\/p>\n<p style=\"text-align: justify;\">\u0422\u043e\u0433\u0434\u0430 \u0432\u044b\u043f\u043e\u043b\u043d\u044f\u044e\u0442\u0441\u044f \u0441\u043b\u0435\u0434\u0443\u044e\u0449\u0438\u0435 \u0441\u0432\u043e\u0439\u0441\u0442\u0432\u0430:<\/p>\n<h4>\u041f\u0440\u0435\u0434\u0435\u043b \u0441\u0443\u043c\u043c\u044b \u0438 \u0440\u0430\u0437\u043d\u043e\u0441\u0442\u0438 \u0444\u0443\u043d\u043a\u0446\u0438\u0439<\/h4>\n<p style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to x_0} \\left(\\alpha f(x) \\pm \\beta g(x) \\right) = \\alpha L \\pm \\beta M<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000080;\"><strong>\u0414\u043e\u043a\u0430\u0437\u0430\u0442\u0435\u043b\u044c\u0441\u0442\u0432\u043e:<\/strong><\/p>\n<p style=\"text-align: justify; \"><a href=\"https:\/\/www.youtube.com\/watch?v=FEPfoAfPsFY&amp;t=2053s\" rel=\"noopener\" target=\"_blank\"><strong>\u0420\u0430\u0441\u0441\u043c\u043e\u0442\u0440\u0438\u043c \u043d\u0430\u0431\u043e\u0440 \u043f\u0440\u0435\u0434\u043f\u043e\u0441\u044b\u043b\u043e\u043a<\/strong><\/a> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle\\mathcal{H}=\\left\\{\\lim_{x\\to x_0} f(x) = L, \\lim_{x\\to x_0} g(x) = M \\right\\}<\/span><\/span>, \u0442\u043e\u0433\u0434\u0430 \u0438\u0441\u0445\u043e\u0434\u044f \u0438\u0437 \u044d\u0442\u043e\u0433\u043e \u043c\u044b \u043c\u043e\u0436\u0435\u043c \u0441\u0434\u0435\u043b\u0430\u0442\u044c \u0441\u043b\u0435\u0434\u0443\u044e\u0449\u0438\u0439 \u0432\u044b\u0432\u043e\u0434:<\/p>\n<table>\n<tbody>\n<tr>\n<td style=\"text-align: right;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(1)<\/span><\/span><\/td>\n<td style=\"text-align: left;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\mathcal{H}\\vdash \\lim_{x\\to x_0}f(x) = L <\/span><\/span>; \u041f\u0440\u0435\u0434\u043f\u043e\u043b\u043e\u0436\u0435\u043d\u0438\u0435<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td style=\"text-align: left;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\mathcal{H}\\vdash \\left(\\forall \\epsilon \\gt 0 \\right)\\left(\\exists \\delta \\gt 0 \\right) \\left(0 \\lt |x-x_0|\\lt \\delta \\rightarrow |f(x) - L|\\lt \\epsilon \\right) <\/span><\/span><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td style=\"text-align: left;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\mathcal{H}\\vdash \\left(\\forall \\epsilon \\gt 0 \\right)\\left(\\exists \\delta \\gt 0 \\right) \\left(0 \\lt |x-x_0|\\lt \\delta \\rightarrow |\\alpha||f(x) - L|\\lt |\\alpha|\\epsilon \\right) <\/span><\/span><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td style=\"text-align: left;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\mathcal{H}\\vdash \\left(\\forall \\epsilon \\gt 0 \\right)\\left(\\exists \\delta \\gt 0 \\right) \\left( 0 \\lt|x-x_0|\\lt \\delta \\rightarrow |\\alpha f(x) - \\alpha L|\\lt |\\alpha|\\epsilon \\right) <\/span><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: right;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(2)<\/span><\/span><\/td>\n<td style=\"text-align: left;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\mathcal{H}\\vdash \\overline{\\epsilon}:= |\\alpha|\\epsilon <\/span><\/span>; \u041e\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u0438\u0435<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: right;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(3)<\/span><\/span><\/td>\n<td style=\"text-align: left;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\mathcal{H}\\vdash \\left(\\forall \\overline{\\epsilon} \\gt 0 \\right)\\left(\\exists \\delta \\gt 0 \\right) \\left(0 \\lt |x-x_0|\\lt \\delta \\rightarrow |\\alpha f(x) - \\alpha L|\\lt \\overline{\\epsilon} \\right) <\/span><\/span>; \u0418\u0437(1,2)<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td style=\"text-align: left;\"><span style=\"background-color: #ffff80; color:#000000\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\mathcal{H}\\vdash \\lim_{x\\to x_0}\\alpha f(x) = \\alpha L <\/span><\/span><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: right;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(4)<\/span><\/span><\/td>\n<td style=\"text-align: left;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\mathcal{H}\\vdash \\lim_{x\\to x_0}g(x) = M <\/span><\/span>; \u041f\u0440\u0435\u0434\u043f\u043e\u043b\u043e\u0436\u0435\u043d\u0438\u0435<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: right;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(5)<\/span><\/span><\/td>\n<td style=\"text-align: left;\"><span style=\"background-color: #ffff80; color:#000000\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\mathcal{H}\\vdash \\lim_{x\\to x_0}\\beta g(x) = \\beta M <\/span><\/span><\/span>; \u0410\u043d\u0430\u043b\u043e\u0433\u0438\u0447\u043d\u043e (3)<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td style=\"text-align: left;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\mathcal{H}\\vdash \\left(\\forall \\overline{\\overline{\\epsilon}} \\gt 0 \\right)\\left(\\exists \\delta \\gt 0 \\right) \\left( 0 \\lt |x-x_0|\\lt \\delta \\rightarrow |\\beta g(x) - \\beta M|\\lt \\overline{\\overline{\\epsilon}} \\right) <\/span><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: right;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(6)<\/span><\/span><\/td>\n<td style=\"text-align: left;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\mathcal{H}\\vdash \\left(\\forall \\overline{\\epsilon},\\overline{\\overline{\\epsilon}} \\gt 0 \\right)\\left(\\exists \\delta \\gt 0 \\right) \\left(0 \\lt |x-x_0|\\lt \\delta \\rightarrow \\left[|\\alpha f(x) - \\alpha L|+ |\\beta g(x) - \\beta M|\\lt \\overline{\\epsilon}+ \\overline{\\overline{\\epsilon}} \\right] \\right) <\/span><\/span>; \u0418\u0437(3,5)<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: right;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(7)<\/span><\/span><\/td>\n<td style=\"text-align: left;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\mathcal{H}\\vdash |\\alpha f(x) - \\alpha L + \\beta g(x) - \\beta M| \\leq |\\alpha f(x) - \\alpha L|+ |\\beta g(x) - \\beta M| <\/span><\/span>; \u041d\u0435\u0440\u0430\u0432\u0435\u043d\u0441\u0442\u0432\u043e \u0422\u0440\u0435\u0443\u0433\u043e\u043b\u044c\u043d\u0438\u043a\u0430: <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(\\forall x,y\\in\\mathbb{R})(|x+y|\\leq |x|+|y|)<\/span><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: right;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(8)<\/span><\/span><\/td>\n<td style=\"text-align: left;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\mathcal{H}\\vdash \\left(\\forall \\overline{\\epsilon},\\overline{\\overline{\\epsilon}} \\gt 0 \\right)\\left(\\exists \\delta \\gt 0 \\right) \\left(0 \\lt |x-x_0|\\lt \\delta \\rightarrow |\\alpha f(x) - \\alpha L + \\beta g(x) - \\beta M| \\lt \\overline{\\epsilon}+ \\overline{\\overline{\\epsilon}} \\right) <\/span><\/span>; \u0418\u0437(6,7)<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: right;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(9)<\/span><\/span><\/td>\n<td style=\"text-align: left;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\epsilon^* := \\overline{\\epsilon} + \\overline{\\overline{\\epsilon}}<\/span><\/span>; \u041e\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u0438\u0435<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: right;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(10)<\/span><\/span><\/td>\n<td style=\"text-align: left;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\mathcal{H}\\vdash \\left(\\forall \\epsilon^* \\gt 0 \\right)\\left(\\exists \\delta \\gt 0 \\right) \\left(0 \\lt |x-x_0|\\lt \\delta \\rightarrow |\\alpha f(x) + \\beta g(x) - \\alpha L - \\beta M| \\lt \\epsilon^* \\right) <\/span><\/span>; \u0418\u0437(8,9)<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td style=\"text-align: left;\"><span style=\"background-color: #ffff80; color:#000000\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\mathcal{H}\\vdash \\lim_{x\\to x_0} (\\alpha f(x) + \\beta g(x)) = \\alpha L + \\beta M <\/span><\/span><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: right;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(11)<\/span><\/span><\/td>\n<td style=\"text-align: left;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\gamma:= - \\beta<\/span><\/span>; \u041e\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u0438\u0435<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: right;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(12)<\/span><\/span><\/td>\n<td style=\"text-align: left;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\mathcal{H}\\vdash \\lim_{x\\to x_0} (\\alpha f(x) + \\gamma g(x)) = \\alpha L + \\gamma M <\/span><\/span>; \u0410\u043d\u0430\u043b\u043e\u0433\u0438\u044f(10)<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: right;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(13)<\/span><\/span><\/td>\n<td style=\"text-align: left;\"><span style=\"background-color: #ffff80; color:#000000\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\mathcal{H}\\vdash \\lim_{x\\to x_0} (\\alpha f(x) - \\beta g(x)) = \\alpha L - \\beta M <\/span><\/span><\/span>; \u0418\u0437(11,12)<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: right;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(14)<\/span><\/span><\/td>\n<td style=\"text-align: left;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\mathcal{H}\\vdash \\lim_{x\\to x_0} (\\alpha f(x) \\pm \\beta g(x)) = \\alpha L \\pm \\beta M <\/span><\/span>; \u0418\u0437(10,13) <\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h4>\u041f\u0440\u0435\u0434\u0435\u043b \u043f\u0440\u043e\u0438\u0437\u0432\u0435\u0434\u0435\u043d\u0438\u044f \u0444\u0443\u043d\u043a\u0446\u0438\u0439<\/h4>\n<p style=\"text-align: center; \"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to x_0} \\left( f(x) g(x) \\right) = L M<\/span><\/span><\/p>\n<p style=\"text-align: justify; \"><a href=\"https:\/\/www.youtube.com\/watch?v=FEPfoAfPsFY&amp;t=2611s\" rel=\"noopener\" target=\"_blank\"><strong>\u042d\u0442\u043e \u0434\u043e\u043a\u0430\u0437\u0430\u0442\u0435\u043b\u044c\u0441\u0442\u0432\u043e \u043d\u0435\u043c\u043d\u043e\u0433\u043e \u0441\u043b\u043e\u0436\u043d\u0435\u0435 \u043f\u0440\u0435\u0434\u044b\u0434\u0443\u0449\u0435\u0433\u043e,<\/strong><\/a> \u043d\u043e \u044d\u0442\u043e \u043d\u0435 \u043f\u0440\u043e\u0431\u043b\u0435\u043c\u0430, \u043a\u043e\u0442\u043e\u0440\u0443\u044e \u043c\u044b \u043d\u0435 \u0441\u043c\u043e\u0436\u0435\u043c \u0440\u0435\u0448\u0438\u0442\u044c \u0441 \u043f\u043e\u043c\u043e\u0449\u044c\u044e \u043d\u0435\u0441\u043a\u043e\u043b\u044c\u043a\u0438\u0445 \u0434\u0440\u0430\u043a\u043e\u043d\u043e\u0432\u0441\u043a\u0438\u0445 \u0442\u0440\u044e\u043a\u043e\u0432. \u0418\u0441\u043f\u043e\u043b\u044c\u0437\u0443\u044f \u0442\u043e\u0442 \u0436\u0435 \u043d\u0430\u0431\u043e\u0440 \u043f\u0440\u0435\u0434\u043f\u043e\u0441\u044b\u043b\u043e\u043a <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathcal{H}<\/span><\/span>, \u0447\u0442\u043e \u0438 \u0432 \u043f\u0440\u0435\u0434\u044b\u0434\u0443\u0449\u0435\u043c \u0434\u043e\u043a\u0430\u0437\u0430\u0442\u0435\u043b\u044c\u0441\u0442\u0432\u0435, \u043c\u044b \u043c\u043e\u0436\u0435\u043c \u043f\u043e\u0441\u0442\u0440\u043e\u0438\u0442\u044c \u0441\u043b\u0435\u0434\u0443\u044e\u0449\u0443\u044e \u0446\u0435\u043f\u043e\u0447\u043a\u0443 \u0440\u0430\u0441\u0441\u0443\u0436\u0434\u0435\u043d\u0438\u0439:<\/p>\n<table>\n<tbody>\n<tr>\n<td style=\"text-align: right;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(1)<\/span><\/span><\/td>\n<td style=\"text-align: left;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\mathcal{H}\\vdash \\overline{\\epsilon} := \\frac{|\\epsilon|}{2(|M|+1)} \\leq \\frac{|\\epsilon|}{2} <\/span><\/span>; \u041e\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u0438\u0435<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: right;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(2)<\/span><\/span><\/td>\n<td style=\"text-align: left;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\mathcal{H}\\vdash \\lim_{x\\to x_0} f(x) = L <\/span><\/span>; \u041f\u0440\u0435\u0434\u043f\u043e\u043b\u043e\u0436\u0435\u043d\u0438\u0435<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td style=\"text-align: left;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\mathcal{H}\\vdash \\left(\\forall \\overline{\\epsilon} \\gt 0 \\right)\\left(\\exists \\delta \\gt 0 \\right)\\left(0 \\lt |x-x_0|\\lt \\delta \\rightarrow |f(x) - L| \\lt \\overline{\\epsilon} = \\frac{|\\epsilon|}{2(|M|+1)}\\right) <\/span><\/span>; \u0418\u0441\u043f\u043e\u043b\u044c\u0437\u0443\u044f (1)<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: right;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(3)<\/span><\/span><\/td>\n<td style=\"text-align: left;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\mathcal{H}\\vdash \\overline{\\overline{\\epsilon}} := \\frac{|\\epsilon|}{2(|L|+1)} \\leq \\frac{|\\epsilon|}{2}<\/span><\/span>; \u041e\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u0438\u0435<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: right;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(4)<\/span><\/span><\/td>\n<td style=\"text-align: left;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\mathcal{H}\\vdash \\lim_{x\\to x_0} g(x) = M <\/span><\/span>; \u041f\u0440\u0435\u0434\u043f\u043e\u043b\u043e\u0436\u0435\u043d\u0438\u0435<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td style=\"text-align: left;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\mathcal{H}\\vdash \\left(\\forall \\overline{\\overline{\\epsilon}} \\gt 0 \\right)\\left(\\exists \\delta \\gt 0 \\right)\\left(0 \\lt |x-x_0|\\lt \\delta \\rightarrow |g(x) - M| \\lt \\overline{\\overline{\\epsilon}} = \\frac{|\\epsilon|}{2(|L|+1)}\\right) <\/span><\/span>; \u0418\u0441\u043f\u043e\u043b\u044c\u0437\u0443\u044f (3)<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: right;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(5)<\/span><\/span><\/td>\n<td style=\"text-align: left;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\mathcal{H}\\vdash |f(x)| - |L| \\lt<\/span><\/span> <span style=\"background-color: #a0ffff; color:#000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">|f(x) - L| \\lt \\overline{\\epsilon} \\lt 1 <\/span><\/span><\/span>; \u041d\u0435\u0440\u0430\u0432\u0435\u043d\u0441\u0442\u0432\u043e \u0422\u0440\u0435\u0443\u0433\u043e\u043b\u044c\u043d\u0438\u043a\u0430 + \u0427\u0430\u0441\u0442\u043d\u044b\u0439 \u0441\u043b\u0443\u0447\u0430\u0439 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\overline{\\epsilon}<\/span><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: right;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(6)<\/span><\/span><\/td>\n<td style=\"text-align: left;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\mathcal{H}\\vdash |f(x)|\\lt 1 + |L| <\/span><\/span>; \u0418\u0437(5)<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: right;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(7)<\/span><\/span><\/td>\n<td style=\"text-align: left;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\mathcal{H}\\vdash |g(x)| - |M| \\lt |g(x) - M| \\lt \\overline{\\overline{\\epsilon}} \\lt 1 <\/span><\/span>; \u041d\u0435\u0440\u0430\u0432\u0435\u043d\u0441\u0442\u0432\u043e \u0422\u0440\u0435\u0443\u0433\u043e\u043b\u044c\u043d\u0438\u043a\u0430 + \u0427\u0430\u0441\u0442\u043d\u044b\u0439 \u0441\u043b\u0443\u0447\u0430\u0439 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\overline{\\overline{\\epsilon}}<\/span><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: right;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(8)<\/span><\/span><\/td>\n<td style=\"text-align: left;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\mathcal{H}\\vdash |g(x)| \\lt 1 + |M| <\/span><\/span>; \u0418\u0437(7)<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: right;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(9)<\/span><\/span><\/td>\n<td style=\"text-align: left;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\mathcal{H}\\vdash |f(x)g(x) - LM|=|<\/span><\/span> <span style=\"background-color: #a0ffa0; color:#000000\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f(x)g(x) - Mf(x)<\/span><\/span><\/span> <span style=\"background-color: #a0a0ff; color:#000000\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">+ Mf(x) - LM<\/span><\/span><\/span> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">|<\/span><\/span>; \u0414\u043e\u0431\u0430\u0432\u0438\u0442\u044c \u043d\u043e\u043b\u044c<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td style=\"text-align: left;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\mathcal{H}\\vdash |f(x)g(x) - LM|=|<\/span><\/span> <span style=\"background-color: #a0ffa0; color:#000000\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f(x)(g(x) - M)<\/span><\/span><\/span> <span style=\"background-color: #a0a0ff; color:#000000\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">+ M (f(x) - L)<\/span><\/span><\/span> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">|<\/span><\/span>; \u0424\u0430\u043a\u0442\u043e\u0440\u0438\u0437\u0430\u0446\u0438\u044f<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: right;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(10)<\/span><\/span><\/td>\n<td style=\"text-align: left;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\mathcal{H}\\vdash |f(x)g(x) - LM|\\leq |<\/span><\/span> <span style=\"background-color: #a0ffa0; color:#000000\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f(x)(g(x) - M)<\/span><\/span><\/span><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">| + |<\/span><\/span> <span style=\"background-color: #a0a0ff; color:#000000\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">M (f(x) - L)<\/span><\/span><\/span> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">|<\/span><\/span>; \u041d\u0435\u0440\u0430\u0432\u0435\u043d\u0441\u0442\u0432\u043e \u0422\u0440\u0435\u0443\u0433\u043e\u043b\u044c\u043d\u0438\u043a\u0430(9)<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td style=\"text-align: left;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\mathcal{H}\\vdash |f(x)g(x) - LM|\\leq <\/span><\/span> <span style=\"background-color: #a0ffa0; color:#000000\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">|f(x)||g(x) - M|<\/span><\/span><\/span><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\"> + <\/span><\/span> <span style=\"background-color: #a0a0ff; color:#000000\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">|M| |f(x) - L|<\/span><\/span><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: right;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(11)<\/span><\/span><\/td>\n<td style=\"text-align: left;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\mathcal{H}\\vdash |f(x)g(x) - LM|\\lt <\/span><\/span> <span style=\"background-color: #a0ffa0; color:#000000\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(1 + |L|)|g(x) - M|<\/span><\/span><\/span><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">+<\/span><\/span> <span style=\"background-color: #a0ffff;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">|M|\\overline{\\epsilon}<\/span><\/span><\/span>; \u0418\u0437(5,6,10)<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: right;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(12)<\/span><\/span><\/td>\n<td style=\"text-align: left;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\mathcal{H}\\vdash \\left[ |g(x) - M|\\lt \\overline{\\overline{\\epsilon}} \\right] \\rightarrow \\left[ (1+|L|)|g(x) - M| + |M|\\overline{\\epsilon} \\lt (1+|L|)\\overline{\\overline{\\epsilon}} + |M|\\overline{\\epsilon}\\right]<\/span><\/span>; \u0418\u0437(11)<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: right;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(13)<\/span><\/span><\/td>\n<td style=\"text-align: left;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\mathcal{H}\\vdash \\left[ |g(x) - M|\\lt \\overline{\\overline{\\epsilon}} \\right] \\rightarrow \\left[ (1+|L|)|g(x) - M| + |M|\\overline{\\epsilon} \\lt (1+|L|)\\frac{|\\epsilon|}{2(|L|+1)} + |M|\\frac{|\\epsilon|}{2(|M|+1)}\\right]<\/span><\/span>; \u0418\u0437(1,3,12)<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td style=\"text-align: left;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\mathcal{H}\\vdash \\left[ |g(x) - M|\\lt \\overline{\\overline{\\epsilon}} \\right] \\rightarrow \\left[ (1+|L|)|g(x) - M| + |M|\\overline{\\epsilon} \\lt \\frac{|\\epsilon|}{2} + \\frac{|\\epsilon||M|}{2(|M|+1)} \\lt \\frac{|\\epsilon|}{2}+ \\frac{|\\epsilon|}{2} = |\\epsilon| \\right]<\/span><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: right;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(14)<\/span><\/span><\/td>\n<td style=\"text-align: left;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\mathcal{H}\\vdash \\left[ |g(x) - M|\\lt \\overline{\\overline{\\epsilon}} \\right] \\rightarrow \\left[ |f(x)g(x) - LM|\\lt |\\epsilon| \\right]<\/span><\/span>; \u0418\u0437(11,13)<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: right;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(15)<\/span><\/span><\/td>\n<td style=\"text-align: left;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\mathcal{H}\\vdash (\\forall \\epsilon \\gt 0 ) (\\exists \\delta \\gt 0 ) \\left(0 \\lt |x-x_0|\\lt \\delta \\rightarrow |f(x)g(x) - LM|\\lt |\\epsilon| \\leq \\epsilon \\right) <\/span><\/span>; \u0418\u0437(1,2,4,14)<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td style=\"text-align: left;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\mathcal{H}\\vdash \\lim_{x\\to x_0}f(x)g(x) = LM.<\/span><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h4>\u041f\u0440\u0435\u0434\u0435\u043b \u043f\u043e\u0441\u0442\u043e\u044f\u043d\u043d\u043e\u0439 \u0444\u0443\u043d\u043a\u0446\u0438\u0438<\/h4>\n<p style=\"text-align: justify; \"><a href=\"https:\/\/www.youtube.com\/watch?v=FEPfoAfPsFY&amp;t=3450s\" rel=\"noopener\" target=\"_blank\"><strong>\u041f\u0440\u0435\u0434\u0435\u043b \u043f\u043e\u0441\u0442\u043e\u044f\u043d\u043d\u043e\u0439 \u0444\u0443\u043d\u043a\u0446\u0438\u0438<\/strong><\/a> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f(x)=c<\/span><\/span>, \u044d\u0442\u043e \u043a\u043e\u043d\u0441\u0442\u0430\u043d\u0442\u0430 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">c<\/span><\/span>. \u0422\u043e \u0435\u0441\u0442\u044c<\/p>\n<p style=\"text-align: center; \"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to x_0}c = c<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000080;\"><strong>\u0414\u043e\u043a\u0430\u0437\u0430\u0442\u0435\u043b\u044c\u0441\u0442\u0432\u043e<\/strong><\/p>\n<p style=\"text-align: justify; \">\u0414\u043e\u043a\u0430\u0437\u0430\u0442\u0435\u043b\u044c\u0441\u0442\u0432\u043e \u044d\u0442\u043e\u0433\u043e \u043d\u0430 \u0441\u0430\u043c\u043e\u043c \u0434\u0435\u043b\u0435 \u043f\u0440\u043e\u0441\u0442\u043e\u0435, \u043f\u043e\u0442\u043e\u043c\u0443 \u0447\u0442\u043e \u044d\u0442\u043e \u0442\u0430\u0432\u0442\u043e\u043b\u043e\u0433\u0438\u044f. \u0423\u0436\u0435 \u0438\u0437\u0432\u0435\u0441\u0442\u043d\u043e, \u0447\u0442\u043e:<\/p>\n<p style=\"text-align: center; \"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to x_0}c = c := (\\forall\\epsilon\\gt 0) (\\exists \\delta \\gt 0)(0\\lt|x-x_0|\\lt \\delta \\rightarrow |c-c|\\lt \\epsilon)<\/span><\/span><\/p>\n<p style=\"text-align: justify; \">\u041d\u043e \u0434\u0435\u043b\u043e \u0432 \u0442\u043e\u043c, \u0447\u0442\u043e <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0=|c-c|\\lt \\epsilon<\/span><\/span> \u2014 \u044d\u0442\u043e \u0442\u0430\u0432\u0442\u043e\u043b\u043e\u0433\u0438\u044f \u0434\u043b\u044f \u043b\u044e\u0431\u043e\u0433\u043e \u043f\u043e\u043b\u043e\u0436\u0438\u0442\u0435\u043b\u044c\u043d\u043e\u0433\u043e \u044d\u043f\u0441\u0438\u043b\u043e\u043d, \u0442\u0430\u043a \u0447\u0442\u043e \u0438\u043c\u043f\u043b\u0438\u043a\u0430\u0446\u0438\u044f \u0442\u0430\u043a\u0436\u0435 \u044f\u0432\u043b\u044f\u0435\u0442\u0441\u044f \u0442\u0430\u0432\u0442\u043e\u043b\u043e\u0433\u0438\u0435\u0439, \u0438, \u0441\u043b\u0435\u0434\u043e\u0432\u0430\u0442\u0435\u043b\u044c\u043d\u043e, \u0432\u044b\u0440\u0430\u0436\u0435\u043d\u0438\u0435 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to x_0}c = c <\/span><\/span> \u0442\u0430\u043a\u0436\u0435 \u044f\u0432\u043b\u044f\u0435\u0442\u0441\u044f \u0442\u0430\u0432\u0442\u043e\u043b\u043e\u0433\u0438\u0435\u0439.<\/p>\n<h4>\u041f\u0440\u0435\u0434\u0435\u043b \u0447\u0430\u0441\u0442\u043d\u043e\u0433\u043e \u0444\u0443\u043d\u043a\u0446\u0438\u0439<\/h4>\n<p style=\"text-align: justify; \"><a href=\"https:\/\/www.youtube.com\/watch?v=FEPfoAfPsFY&amp;t=3563s\" rel=\"noopener\" target=\"_blank\"><strong>\u0422\u0435\u043f\u0435\u0440\u044c \u043c\u044b \u043c\u043e\u0436\u0435\u043c \u0434\u043e\u043a\u0430\u0437\u0430\u0442\u044c \u043f\u0440\u0430\u0432\u0438\u043b\u043e \u0434\u043b\u044f \u043f\u0440\u0435\u0434\u0435\u043b\u0430 \u0447\u0430\u0441\u0442\u043d\u043e\u0433\u043e \u0434\u0432\u0443\u0445 \u0444\u0443\u043d\u043a\u0446\u0438\u0439.<\/strong><\/a> \u042d\u0442\u043e<\/p>\n<p style=\"text-align: center; \"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to x_0}\\frac{f(x)}{g(x)}= \\frac{L}{M}<\/span><\/span><\/p>\n<p style=\"text-align: justify; \">\u0413\u0434\u0435, \u043a\u0430\u043a \u0438 \u0432 \u043f\u0440\u0435\u0434\u044b\u0434\u0443\u0449\u0438\u0445 \u0441\u0432\u043e\u0439\u0441\u0442\u0432\u0430\u0445, \u043c\u044b \u043f\u0440\u0435\u0434\u043f\u043e\u043b\u0430\u0433\u0430\u0435\u043c, \u0447\u0442\u043e \u0432\u044b\u043f\u043e\u043b\u043d\u044f\u0435\u0442\u0441\u044f \u043d\u0430\u0431\u043e\u0440 \u043f\u0440\u0435\u0434\u043f\u043e\u0441\u044b\u043b\u043e\u043a<\/p>\n<p style=\"text-align: center; \"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\mathcal{H}=\\{\\lim_{x\\to x_0}f(x) = L, \\lim_{x\\to x_0}g(x) = M\\}<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000080;\"><strong>\u0414\u043e\u043a\u0430\u0437\u0430\u0442\u0435\u043b\u044c\u0441\u0442\u0432\u043e<\/strong><\/p>\n<p style=\"text-align: justify; \">\u041a \u0441\u0447\u0430\u0441\u0442\u044c\u044e, \u043d\u0430\u043c \u0431\u043e\u043b\u044c\u0448\u0435 \u043d\u0435 \u043f\u0440\u0438\u0434\u0435\u0442\u0441\u044f \u043f\u0440\u043e\u0432\u043e\u0434\u0438\u0442\u044c \u0442\u0430\u043a\u0438\u0435 \u0434\u043e\u043a\u0430\u0437\u0430\u0442\u0435\u043b\u044c\u0441\u0442\u0432\u0430, \u043a\u0430\u043a \u0440\u0430\u043d\u044c\u0448\u0435, \u043f\u043e\u0442\u043e\u043c\u0443 \u0447\u0442\u043e \u0442\u0435\u043f\u0435\u0440\u044c \u043c\u044b \u043c\u043e\u0436\u0435\u043c \u043d\u0430\u043f\u0440\u044f\u043c\u0443\u044e \u0438\u0441\u043f\u043e\u043b\u044c\u0437\u043e\u0432\u0430\u0442\u044c \u044d\u0442\u0438 \u0440\u0435\u0437\u0443\u043b\u044c\u0442\u0430\u0442\u044b \u0434\u043b\u044f \u0434\u043e\u0441\u0442\u0438\u0436\u0435\u043d\u0438\u044f \u043d\u0430\u0448\u0438\u0445 \u0446\u0435\u043b\u0435\u0439. \u041d\u043e \u043f\u0435\u0440\u0435\u0434 \u044d\u0442\u0438\u043c \u0441\u043d\u0430\u0447\u0430\u043b\u0430 \u0434\u043e\u043a\u0430\u0436\u0435\u043c, \u0447\u0442\u043e<\/p>\n<p style=\"text-align: center; \"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to x_0}\\frac{1}{g(x)} = \\frac{1}{M}<\/span><\/span><\/p>\n<p style=\"text-align: justify; \">\u0427\u0442\u043e\u0431\u044b \u0434\u043e\u043a\u0430\u0437\u0430\u0442\u044c \u044d\u0442\u043e, \u0434\u043e\u0441\u0442\u0430\u0442\u043e\u0447\u043d\u043e \u0438\u0441\u043f\u043e\u043b\u044c\u0437\u043e\u0432\u0430\u0442\u044c \u043f\u0440\u0430\u0432\u0438\u043b\u043e \u043f\u0440\u0435\u0434\u0435\u043b\u0430 \u043f\u0440\u043e\u0438\u0437\u0432\u0435\u0434\u0435\u043d\u0438\u044f \u0438 \u043f\u0440\u0435\u0434\u0435\u043b\u0430 \u043f\u043e\u0441\u0442\u043e\u044f\u043d\u043d\u043e\u0439 \u0444\u0443\u043d\u043a\u0446\u0438\u0438 \u0432 \u0441\u043e\u0447\u0435\u0442\u0430\u043d\u0438\u0438, \u043d\u0430\u043c \u043f\u0440\u043e\u0441\u0442\u043e \u043d\u0443\u0436\u043d\u043e \u0443\u0431\u0435\u0434\u0438\u0442\u044c\u0441\u044f, \u0447\u0442\u043e <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">g(x)<\/span><\/span> \u043d\u0435 \u0440\u0430\u0432\u043d\u043e \u043d\u0443\u043b\u044e:<\/p>\n<p style=\"text-align: center; \"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle 1 = \\lim_{x\\to x_0}\\left( 1 \\right) \\lim_{x\\to x_0}\\left( g(x) \\cdot \\frac{1}{g(x)} \\right) = \\lim_{x\\to x_0}g(x) \\cdot \\lim_{x\\to x_0} \\frac{1}{g(x)} = M \\cdot \\lim_{x\\to x_0} \\frac{1}{g(x)}<\/span><\/span><\/p>\n<p style=\"text-align: center; \">\u0421\u043b\u0435\u0434\u043e\u0432\u0430\u0442\u0435\u043b\u044c\u043d\u043e: <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to x_0} \\frac{1}{g(x)} = \\frac{1}{M}<\/span><\/span><\/p>\n<p style=\"text-align: justify; \">\u041d\u0430\u043a\u043e\u043d\u0435\u0446, \u043f\u043e \u043f\u0440\u0430\u0432\u0438\u043b\u0443 \u043f\u0440\u0435\u0434\u0435\u043b\u0430 \u043f\u0440\u043e\u0438\u0437\u0432\u0435\u0434\u0435\u043d\u0438\u044f \u0438\u043c\u0435\u0435\u043c:<\/p>\n<p style=\"text-align: center; \"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to x_0} \\frac{f(x)}{g(x)} = \\lim_{x\\to x_0} f(x) \\frac{1}{g(x)}= L \\cdot\\frac{1}{M} = \\frac{L}{M}<\/span><\/span><\/p>\n<p style=\"text-align: justify; \">\u042d\u0442\u043e \u0441\u043f\u0440\u0430\u0432\u0435\u0434\u043b\u0438\u0432\u043e, \u043f\u043e\u043a\u0430 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">M<\/span><\/span> \u043d\u0435 \u0440\u0430\u0432\u043d\u043e \u043d\u0443\u043b\u044e.<\/p>\n<h4>\u041f\u0440\u0435\u0434\u0435\u043b \u043d\u0430\u0442\u0443\u0440\u0430\u043b\u044c\u043d\u043e\u0439 \u0441\u0442\u0435\u043f\u0435\u043d\u0438<\/h4>\n<p style=\"text-align: justify; \"><a href=\"https:\/\/www.youtube.com\/watch?v=FEPfoAfPsFY&amp;t=3725s\" rel=\"noopener\" target=\"_blank\"><strong>\u042d\u0442\u043e \u0441\u0432\u043e\u0439\u0441\u0442\u0432\u043e \u0433\u043e\u0432\u043e\u0440\u0438\u0442 \u043d\u0430\u043c, \u0447\u0442\u043e,<\/strong><\/a> \u0435\u0441\u043b\u0438 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x_0 \\to x_0}f(x) = L<\/span><\/span>, \u0442\u043e \u0432\u044b\u043f\u043e\u043b\u043d\u044f\u0435\u0442\u0441\u044f <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\left(\\forall n \\in \\mathbb{N}\\right) \\left( \\lim_{x\\to x_0} \\left( [f(x)]^n \\right) = L^n \\right)<\/span><\/span>. \u042d\u0442\u043e \u043c\u043e\u0436\u043d\u043e \u0434\u043e\u043a\u0430\u0437\u0430\u0442\u044c \u0441 \u043f\u043e\u043c\u043e\u0449\u044c\u044e \u043c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u0447\u0435\u0441\u043a\u043e\u0439 \u0438\u043d\u0434\u0443\u043a\u0446\u0438\u0438.<\/p>\n<p style=\"text-align: justify; color: #000080;\"><strong>\u0414\u043e\u043a\u0430\u0437\u0430\u0442\u0435\u043b\u044c\u0441\u0442\u0432\u043e:<\/strong><\/p>\n<ul style=\"text-align: justify; \">\n<li><strong>\u0421\u043b\u0443\u0447\u0430\u0439 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">n=1<\/span><\/span>:<\/strong> (\u043d\u0430\u0447\u0430\u043b\u044c\u043d\u044b\u0439 \u0448\u0430\u0433)\n<p style=\"text-align: justify;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to x_0} [f(x)]^1 = \\lim_{x\\to x_0} f(x) = L.<\/span><\/span> \u042d\u0442\u043e \u0437\u0430\u0432\u0435\u0440\u0448\u0430\u0435\u0442 \u043d\u0430\u0447\u0430\u043b\u044c\u043d\u044b\u0439 \u0448\u0430\u0433 \u2705<\/p>\n<\/li>\n<li><strong>\u0421\u043b\u0443\u0447\u0430\u0439 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">n=k<\/span><\/span>:<\/strong> (\u0438\u043d\u0434\u0443\u043a\u0442\u0438\u0432\u043d\u044b\u0439 \u0448\u0430\u0433)\n<p style=\"text-align: justify;\">\u041f\u0440\u0435\u0434\u043f\u043e\u043b\u0430\u0433\u0430\u044f, \u0447\u0442\u043e \u0432\u044b\u043f\u043e\u043b\u043d\u044f\u0435\u0442\u0441\u044f: <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to x_0} [f(x)]^k = L^k <\/span><\/span> (\u0418\u043d\u0434\u0443\u043a\u0442\u0438\u0432\u043d\u0430\u044f \u0413\u0438\u043f\u043e\u0442\u0435\u0437\u0430), \u043f\u0440\u043e\u0432\u0435\u0440\u0438\u043c, \u0447\u0442\u043e \u0441\u043b\u0435\u0434\u043e\u0432\u0430\u0442\u0435\u043b\u044c\u043d\u043e \u0432\u044b\u043f\u043e\u043b\u043d\u044f\u0435\u0442\u0441\u044f <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to x_0} [f(x)]^{k+1} = L^{k+1} <\/span><\/span>.<\/p>\n<p style=\"text-align: justify;\">\u0418\u043c\u0435\u0435\u043c: <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to x_0} [f(x)]^{k+1} = \\lim_{x\\to x_0} \\{f(x) [f(x)]^k\\} = \\lim_{x\\to x_0}f(x) \\lim_{x\\to x_0} [f(x)]^{k} =L \\lim_{x\\to x_0} [f(x)]^{k}<\/span><\/span>. \u042d\u0442\u043e \u0441\u043b\u0435\u0434\u0443\u0435\u0442 \u0438\u0437 \u0434\u043e\u043a\u0430\u0437\u0430\u043d\u043d\u043e\u0433\u043e \u0432\u044b\u0448\u0435 \u043f\u0440\u0430\u0432\u0438\u043b\u0430 \u043f\u0440\u0435\u0434\u0435\u043b\u0430 \u043f\u0440\u043e\u0438\u0437\u0432\u0435\u0434\u0435\u043d\u0438\u044f.<\/p>\n<p style=\"text-align: justify;\">\u0414\u0430\u043b\u0435\u0435, \u043f\u043e \u0438\u043d\u0434\u0443\u043a\u0442\u0438\u0432\u043d\u043e\u0439 \u0433\u0438\u043f\u043e\u0442\u0435\u0437\u0435 \u043f\u043e\u043b\u0443\u0447\u0430\u0435\u043c <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to x_0} [f(x)]^{k+1} = L \\lim_{x\\to x_0} [f(x)]^{k} =L\\cdot L^k = L^{k+1}.<\/span><\/span> \u042d\u0442\u043e \u0437\u0430\u0432\u0435\u0440\u0448\u0430\u0435\u0442 \u0438\u043d\u0434\u0443\u043a\u0442\u0438\u0432\u043d\u044b\u0439 \u0448\u0430\u0433 \u2705<\/p>\n<\/li>\n<li>\u0421\u043b\u0435\u0434\u043e\u0432\u0430\u0442\u0435\u043b\u044c\u043d\u043e: <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\left(\\forall n \\in \\mathbb{N}\\right) \\left( \\lim_{x\\to x_0} \\left( [f(x)]^n \\right) = L^n \\right). <\/span><\/span><\/li>\n<\/ul>\n<h4>\u041f\u0440\u0435\u0434\u0435\u043b \u043a\u043e\u0440\u043d\u044f n-\u043e\u0439 \u0441\u0442\u0435\u043f\u0435\u043d\u0438<\/h4>\n<p style=\"text-align: justify; \"><a href=\"https:\/\/www.youtube.com\/watch?v=FEPfoAfPsFY&amp;t=3912s\" rel=\"noopener\" target=\"_blank\"><strong>\u0410\u043d\u0430\u043b\u043e\u0433\u0438\u0447\u043d\u043e \u0441\u0442\u0435\u043f\u0435\u043d\u0438, \u0441\u043f\u0440\u0430\u0432\u0435\u0434\u043b\u0438\u0432\u043e, \u0447\u0442\u043e<\/strong><\/a> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\left(\\forall n \\in \\mathbb{\u041d}\\right) \\left( \\lim_{x\\to x_0} \\sqrt[n]{f(x)} = \\sqrt[n]{L} \\right) <\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000080;\"><strong>\u0414\u043e\u043a\u0430\u0437\u0430\u0442\u0435\u043b\u044c\u0441\u0442\u0432\u043e:<\/strong><\/p>\n<p style=\"text-align: justify; \">\u0418\u0441\u043f\u043e\u043b\u044c\u0437\u0443\u044f \u0442\u043e\u043b\u044c\u043a\u043e \u0447\u0442\u043e \u0434\u043e\u043a\u0430\u0437\u0430\u043d\u043d\u043e\u0435 \u043f\u0440\u0430\u0432\u0438\u043b\u043e \u0434\u043b\u044f \u0441\u0442\u0435\u043f\u0435\u043d\u0438, \u0438\u043c\u0435\u0435\u043c<\/p>\n<p style=\"text-align: center; \"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle L= \\lim_{x\\to x_0} f(x)=\\lim_{x\\to x_0} \\left[\\sqrt[n]{f(x)}\\right]^n = \\left[ \\lim_{x\\to x_0} \\sqrt[n]{f(x)}\\right]^n <\/span><\/span><\/p>\n<p style=\"text-align: center; \">\u0421\u043b\u0435\u0434\u043e\u0432\u0430\u0442\u0435\u043b\u044c\u043d\u043e: <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to x_0} \\sqrt[n]{f(x)} =\\sqrt[n]{L}.<\/span><\/span><\/p>\n<h4>\u041f\u0440\u0435\u0434\u0435\u043b \u0434\u0440\u043e\u0431\u043d\u044b\u0445 \u0441\u0442\u0435\u043f\u0435\u043d\u0435\u0439<\/h4>\n<p style=\"text-align: justify; \"><a href=\"https:\/\/www.youtube.com\/watch?v=FEPfoAfPsFY&amp;t=4007s\" rel=\"noopener\" target=\"_blank\"><strong>\u0421 \u043e\u0431\u044a\u0435\u0434\u0438\u043d\u0435\u043d\u0438\u0435\u043c \u043f\u043e\u0441\u043b\u0435\u0434\u043d\u0438\u0445 \u0434\u0432\u0443\u0445 \u0434\u043e\u043a\u0430\u0437\u0430\u0442\u0435\u043b\u044c\u0441\u0442\u0432<\/strong><\/a> \u043c\u044b \u043c\u043e\u0436\u0435\u043c \u0437\u0430\u043a\u043b\u044e\u0447\u0438\u0442\u044c \u043d\u0430\u0448\u0435 \u043f\u043e\u0441\u043b\u0435\u0434\u043d\u0435\u0435 \u0434\u043e\u043a\u0430\u0437\u0430\u0442\u0435\u043b\u044c\u0441\u0442\u0432\u043e, \u044d\u0442\u043e: <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\left(\\forall p,q\\neq 0 \\in \\mathbb{Z}\\right) \\left( \\lim_{x\\to x_0} \\left[f(x)\\right]^{\\frac{p}{q}} = L^{\\frac{p}{q}} \\right). <\/span><\/span>, \u043a\u043e\u0442\u043e\u0440\u043e\u0435 \u043f\u043e\u043b\u0443\u0447\u0430\u0435\u0442\u0441\u044f \u0431\u043b\u0430\u0433\u043e\u0434\u0430\u0440\u044f \u043f\u0440\u0430\u0432\u0438\u043b\u0443 \u043f\u0440\u043e\u0438\u0437\u0432\u0435\u0434\u0435\u043d\u0438\u044f, \u043f\u043e\u0441\u043a\u043e\u043b\u044c\u043a\u0443 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle [f(x)]^{\\frac{p}{q}} =[\\sqrt[q]{f(x)}]^p <\/span><\/span> \u0438 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle L^{\\frac{p}{q}} =[\\sqrt[q]{L}]^p. <\/span><\/span><\/p>\n<h4>\u041f\u0440\u0435\u0434\u0435\u043b <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to x_0}x = x_0<\/span><\/span><\/h4>\n<p style=\"text-align: justify; \"><a href=\"https:\/\/www.youtube.com\/watch?v=FEPfoAfPsFY&amp;t=4073s\" rel=\"noopener\" target=\"_blank\"><strong>\u042d\u0442\u0438\u043c \u0434\u043e\u043a\u0430\u0437\u0430\u0442\u0435\u043b\u044c\u0441\u0442\u0432\u043e\u043c \u0437\u0430\u0432\u0435\u0440\u0448\u0430\u0435\u043c \u044d\u0442\u043e\u0442 \u0446\u0438\u043a\u043b \u0434\u043e\u043a\u0430\u0437\u0430\u0442\u0435\u043b\u044c\u0441\u0442\u0432,<\/strong><\/a> \u0441 \u044d\u0442\u0438\u043c \u0438 \u043f\u0440\u0435\u0434\u044b\u0434\u0443\u0449\u0438\u043c\u0438 \u043c\u044b \u0441\u043c\u043e\u0436\u0435\u043c \u0432 \u0434\u0430\u043b\u044c\u043d\u0435\u0439\u0448\u0435\u043c \u0432\u044b\u0447\u0438\u0441\u043b\u044f\u0442\u044c \u0431\u043e\u043b\u044c\u0448\u043e\u0435 \u043a\u043e\u043b\u0438\u0447\u0435\u0441\u0442\u0432\u043e \u043f\u0440\u0435\u0434\u0435\u043b\u043e\u0432 \u043f\u0440\u0430\u043a\u0442\u0438\u0447\u0435\u0441\u043a\u0438 \u0438\u043d\u0442\u0443\u0438\u0442\u0438\u0432\u043d\u043e.<\/p>\n<p style=\"text-align: justify; \">\u041f\u0440\u043e\u0441\u0442\u043e \u0434\u043e\u043a\u0430\u0437\u0430\u0442\u044c, \u0447\u0442\u043e <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to x_0}x = x_0<\/span><\/span>, \u043f\u043e\u0442\u043e\u043c\u0443 \u0447\u0442\u043e \u0434\u043b\u044f \u044d\u0442\u043e\u0433\u043e \u043d\u0435\u043e\u0431\u0445\u043e\u0434\u0438\u043c\u043e, \u0447\u0442\u043e\u0431\u044b<\/p>\n<p style=\"text-align: center; \"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(\\forall \\epsilon \\gt 0) (\\exists \\delta \\gt 0)(0\\lt |x-x_0|\\lt \\delta\\rightarrow |x-x_0|\\lt \\epsilon)<\/span><\/span><\/p>\n<p style=\"text-align: justify; \">\u0421\u043e\u0433\u043b\u0430\u0441\u043d\u043e \u043e\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u0438\u044e \u043f\u0440\u0435\u0434\u0435\u043b\u0430, \u0434\u043b\u044f \u043b\u044e\u0431\u043e\u0433\u043e \u044d\u043f\u0441\u0438\u043b\u043e\u043d \u0434\u043e\u043b\u0436\u043d\u043e \u0441\u0443\u0449\u0435\u0441\u0442\u0432\u043e\u0432\u0430\u0442\u044c \u0445\u043e\u0442\u044f \u0431\u044b \u043e\u0434\u043d\u043e \u0434\u0435\u043b\u044c\u0442\u0430, \u0434\u043b\u044f \u043a\u043e\u0442\u043e\u0440\u043e\u0433\u043e \u0432\u044b\u043f\u043e\u043b\u043d\u044f\u0435\u0442\u0441\u044f \u0432\u0441\u0435 \u043e\u0441\u0442\u0430\u043b\u044c\u043d\u043e\u0435; \u0442\u0430\u043a \u0447\u0442\u043e \u0434\u043e\u0441\u0442\u0430\u0442\u043e\u0447\u043d\u043e \u043d\u0430\u0439\u0442\u0438 \u043e\u0434\u043d\u043e, \u0447\u0442\u043e\u0431\u044b \u043f\u043e\u0434\u0442\u0432\u0435\u0440\u0434\u0438\u0442\u044c, \u0447\u0442\u043e \u043f\u0440\u0435\u0434\u0435\u043b \u0434\u0435\u0439\u0441\u0442\u0432\u0438\u0442\u0435\u043b\u044c\u043d\u043e \u0440\u0430\u0432\u0435\u043d \u0437\u0430\u044f\u0432\u043b\u0435\u043d\u043d\u043e\u043c\u0443. \u041d\u043e \u044d\u0442\u043e \u043d\u0430 \u0441\u0430\u043c\u043e\u043c \u0434\u0435\u043b\u0435 \u043e\u0447\u0435\u0432\u0438\u0434\u043d\u043e, \u043f\u043e\u0442\u043e\u043c\u0443 \u0447\u0442\u043e \u043b\u044e\u0431\u043e\u0435 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\delta\\leq\\epsilon<\/span><\/span> \u0443\u0434\u043e\u0432\u043b\u0435\u0442\u0432\u043e\u0440\u044f\u0435\u0442 \u044d\u0442\u043e\u043c\u0443 \u0443\u0441\u043b\u043e\u0432\u0438\u044e. \u0421\u043b\u0435\u0434\u043e\u0432\u0430\u0442\u0435\u043b\u044c\u043d\u043e: <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to x_0}x = x_0.<\/span><\/span><\/p>\n<p><a name=\"7\"><\/a><\/p>\n<h2>\u0412\u044b\u0447\u0438\u0441\u043b\u0435\u043d\u0438\u0435 \u043f\u0440\u043e\u0441\u0442\u044b\u0445 \u043f\u0440\u0435\u0434\u0435\u043b\u043e\u0432<\/h2>\n<p style=\"text-align: justify; \"><a href=\"https:\/\/www.youtube.com\/watch?v=FEPfoAfPsFY&amp;t=4155s\" rel=\"noopener\" target=\"_blank\"><strong>\u0411\u043b\u0430\u0433\u043e\u0434\u0430\u0440\u044f \u0432\u0441\u0435\u043c \u044d\u0442\u0438\u043c \u0442\u0435\u043e\u0440\u0435\u043c\u0430\u043c, \u043a\u043e\u0442\u043e\u0440\u044b\u0435 \u043c\u044b \u0442\u043e\u043b\u044c\u043a\u043e \u0447\u0442\u043e \u0440\u0430\u0441\u0441\u043c\u043e\u0442\u0440\u0435\u043b\u0438<\/strong><\/a>, \u043c\u043e\u0436\u043d\u043e \u0432\u044b\u0447\u0438\u0441\u043b\u044f\u0442\u044c \u043c\u043d\u043e\u0436\u0435\u0441\u0442\u0432\u043e \u043f\u0440\u0435\u0434\u0435\u043b\u043e\u0432 \u0434\u043e\u0432\u043e\u043b\u044c\u043d\u043e \u0438\u043d\u0442\u0443\u0438\u0442\u0438\u0432\u043d\u043e, \u043a\u0430\u043a \u0435\u0441\u043b\u0438 \u0431\u044b \u043c\u044b \u043f\u0440\u043e\u0441\u0442\u043e \u043f\u043e\u0434\u0441\u0442\u0430\u0432\u043b\u044f\u043b\u0438 \u0437\u043d\u0430\u0447\u0435\u043d\u0438\u044f \u0432 \u0444\u0443\u043d\u043a\u0446\u0438\u044e. \u0412\u043e\u0442 \u043d\u0435\u0441\u043a\u043e\u043b\u044c\u043a\u043e \u043f\u0440\u0438\u043c\u0435\u0440\u043e\u0432:<\/p>\n<ol style=\"text-align:left; \">\n<li>\n<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">{}\\\\ \\begin{array}{rl}\n\n \\displaystyle \\lim_{x\\to 2}(x^2 + 4x) &amp; = \\displaystyle \\lim_{x\\to 2}(x^2) + \\lim_{x\\to 2}(4x) \\\\ \\\\\n\n&amp; = \\displaystyle \\left(\\lim_{x\\to 2} x \\right)^2 + 4\\lim_{x\\to 2} x \\\\ \\\\\n\n&amp; = (2)^2 + 8 = 12\n\n\\end{array}<\/span><\/span><\/li>\n<li><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">{} \\\\ \\begin{array}{rl}\n\n\\displaystyle \\lim_{x\\to 1}\\left.\\frac{(3x-1)^2}{(x+1)^3} \\right. &amp; = \\displaystyle \\frac{(3(1)-1)^2}{((1)+1)^3} \\\\ \\\\\n\n&amp; = \\displaystyle \\frac{4}{8} = \\frac{1}{2}\n\n\\end{array}\n\n<\/span><\/span><\/li>\n<li><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">{} \\\\ \\begin{array}{rl}\n\n\\displaystyle \\lim_{x\\to 2} \\frac{x-2}{x^2 - 4} &amp;= \\displaystyle \\lim_{x\\to 2} \\frac{x-2}{(x-2)(x+2)} \\\\ \\\\\n\n&amp; = \\displaystyle \\lim_{x\\to 2} \\frac{1}{x+2} = \\dfrac{1}{4}\n\n\\end{array}\n\n <\/span><\/span><\/li>\n<li><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">{} \\\\ \\begin{array}{rl}\n\n\\displaystyle \\lim_{h\\to 0} \\frac{(x+h)^3-x^3}{h} &amp;= \\displaystyle \\lim_{h\\to 0} \\frac{x^3 + 3x^2 h + 3xh^2 -x^3}{h} \\\\ \\\\\n\n&amp; = \\displaystyle\\lim_{h\\to 0} \\frac{3x^3 h + 3xh^2}{h} \\\\ \\\\\n\n&amp; = \\displaystyle \\lim_{h\\to 0} 3x^2 + 3xh = 3x^2\n\n\\end{array}\n\n <\/span><\/span><\/li>\n<li><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">{} \\\\ \\begin{array}{rl}\n\n\\displaystyle \\lim_{x\\to 1} \\frac{x-1}{\\sqrt{x^2 + 3} - 2 } &amp;=\\displaystyle \\lim_{x\\to 1} \\frac{x-1}{\\sqrt{x^2 + 3} - 2 } \\frac{\\sqrt{x^2 + 3} + 2}{\\sqrt{x^2 + 3} + 2} \\\\ \\\\\n\n&amp; =\\displaystyle \\lim_{x\\to 1} \\frac{(x-1)(\\sqrt{x^2 + 3} + 2)}{(x^2 + 3) - 4 } \\\\ \\\\\n\n&amp; =\\displaystyle \\lim_{x\\to 1} \\frac{(x-1)(\\sqrt{x^2 + 3} + 2)}{x^2 -1 } \\\\ \\\\\n\n&amp; =\\displaystyle \\lim_{x\\to 1} \\frac{(x-1)(\\sqrt{x^2 + 3} + 2)}{(x-1)(x+1) } \\\\ \\\\\n\n&amp; =\\displaystyle \\lim_{x\\to 1} \\frac{\\sqrt{x^2 + 3} + 2}{ x+1 } \\\\ \\\\\n\n&amp; =\\displaystyle \\frac{2+2}{2} =2\n\n\\end{array}<\/span><\/span><\/li>\n<\/ol>\n","protected":false},"excerpt":{"rendered":"<p>\u041f\u0440\u0435\u0434\u0435\u043b \u0444\u0443\u043d\u043a\u0446\u0438\u0438 \u043e\u0434\u043d\u043e\u0439 \u0434\u0435\u0439\u0441\u0442\u0432\u0438\u0442\u0435\u043b\u044c\u043d\u043e\u0439 \u043f\u0435\u0440\u0435\u043c\u0435\u043d\u043d\u043e\u0439 \u0420\u0435\u0437\u044e\u043c\u0435: \u041d\u0430 \u044d\u0442\u043e\u043c \u0437\u0430\u043d\u044f\u0442\u0438\u0438 \u043f\u043e\u0434\u0440\u043e\u0431\u043d\u043e \u0440\u0430\u0441\u0441\u043c\u0430\u0442\u0440\u0438\u0432\u0430\u0435\u0442\u0441\u044f \u0444\u043e\u0440\u043c\u0430\u043b\u044c\u043d\u043e\u0435 \u043e\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u0438\u0435 \u043f\u0440\u0435\u0434\u0435\u043b\u0430 \u0444\u0443\u043d\u043a\u0446\u0438\u0438 \u043e\u0434\u043d\u043e\u0439 \u0434\u0435\u0439\u0441\u0442\u0432\u0438\u0442\u0435\u043b\u044c\u043d\u043e\u0439 \u043f\u0435\u0440\u0435\u043c\u0435\u043d\u043d\u043e\u0439, \u0438 \u043d\u0430 \u043e\u0441\u043d\u043e\u0432\u0435 \u044d\u0442\u043e\u0433\u043e \u0434\u0435\u043c\u043e\u043d\u0441\u0442\u0440\u0438\u0440\u0443\u044e\u0442\u0441\u044f \u043e\u0441\u043d\u043e\u0432\u043d\u044b\u0435 \u0441\u0432\u043e\u0439\u0441\u0442\u0432\u0430, \u043a\u043e\u0442\u043e\u0440\u044b\u0435 \u043f\u0440\u0438\u0432\u043e\u0434\u044f\u0442 \u043a \u0430\u043b\u0433\u0435\u0431\u0440\u0435 \u043f\u0440\u0435\u0434\u0435\u043b\u043e\u0432. \u0426\u0435\u043b\u0438 \u043e\u0431\u0443\u0447\u0435\u043d\u0438\u044f: \u041a \u043a\u043e\u043d\u0446\u0443 \u044d\u0442\u043e\u0433\u043e \u0437\u0430\u043d\u044f\u0442\u0438\u044f \u0441\u0442\u0443\u0434\u0435\u043d\u0442 \u0441\u043c\u043e\u0436\u0435\u0442: \u0417\u0430\u043f\u043e\u043c\u043d\u0438\u0442\u044c \u043e\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u0438\u0435 \u043f\u0440\u0435\u0434\u0435\u043b\u0430 \u0444\u0443\u043d\u043a\u0446\u0438\u0438 \u043e\u0434\u043d\u043e\u0439 \u0434\u0435\u0439\u0441\u0442\u0432\u0438\u0442\u0435\u043b\u044c\u043d\u043e\u0439 \u043f\u0435\u0440\u0435\u043c\u0435\u043d\u043d\u043e\u0439. \u0414\u043e\u043a\u0430\u0437\u0430\u0442\u044c \u0441\u0432\u043e\u0439\u0441\u0442\u0432\u0430, \u043a\u043e\u0442\u043e\u0440\u044b\u0435 \u043f\u0440\u0438\u0432\u043e\u0434\u044f\u0442 \u043a \u0430\u043b\u0433\u0435\u0431\u0440\u0435 \u043f\u0440\u0435\u0434\u0435\u043b\u043e\u0432, \u0438\u0441\u043f\u043e\u043b\u044c\u0437\u0443\u044f \u0432\u044b\u0432\u043e\u0434\u044b . 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