{"id":27948,"date":"2024-08-16T13:00:30","date_gmt":"2024-08-16T13:00:30","guid":{"rendered":"http:\/\/toposuranos.com\/material\/?p=27948"},"modified":"2024-08-18T06:55:23","modified_gmt":"2024-08-18T06:55:23","slug":"%d1%82%d0%b5%d0%be%d1%80%d0%b5%d0%bc%d0%b0-%d1%81%d1%8d%d0%bd%d0%b4%d0%b2%d0%b8%d1%87%d0%b0-%d0%b4%d0%bb%d1%8f-%d0%b2%d1%8b%d1%87%d0%b8%d1%81%d0%bb%d0%b5%d0%bd%d0%b8%d1%8f-%d0%bf%d1%80%d0%b5%d0%b4","status":"publish","type":"post","link":"https:\/\/toposuranos.com\/material\/ru\/%d1%82%d0%b5%d0%be%d1%80%d0%b5%d0%bc%d0%b0-%d1%81%d1%8d%d0%bd%d0%b4%d0%b2%d0%b8%d1%87%d0%b0-%d0%b4%d0%bb%d1%8f-%d0%b2%d1%8b%d1%87%d0%b8%d1%81%d0%bb%d0%b5%d0%bd%d0%b8%d1%8f-%d0%bf%d1%80%d0%b5%d0%b4\/","title":{"rendered":"\u0422\u0435\u043e\u0440\u0435\u043c\u0430 \u0421\u044d\u043d\u0434\u0432\u0438\u0447\u0430 \u0434\u043b\u044f \u0432\u044b\u0447\u0438\u0441\u043b\u0435\u043d\u0438\u044f \u043f\u0440\u0435\u0434\u0435\u043b\u043e\u0432"},"content":{"rendered":"<p><center><\/p>\n<h1>\u0422\u0435\u043e\u0440\u0435\u043c\u0430 \u0421\u044d\u043d\u0434\u0432\u0438\u0447\u0430 \u0434\u043b\u044f \u0432\u044b\u0447\u0438\u0441\u043b\u0435\u043d\u0438\u044f \u043f\u0440\u0435\u0434\u0435\u043b\u043e\u0432<\/h1>\n<p><em><strong>\u0420\u0435\u0437\u044e\u043c\u0435:<\/strong><br \/>\n\u042d\u0442\u043e\u0442 \u0443\u0440\u043e\u043a \u043f\u0440\u0435\u0434\u0441\u0442\u0430\u0432\u043b\u044f\u0435\u0442 \u0422\u0435\u043e\u0440\u0435\u043c\u0443 \u0421\u044d\u043d\u0434\u0432\u0438\u0447\u0430, \u043a\u043b\u044e\u0447\u0435\u0432\u043e\u0439 \u0438\u043d\u0441\u0442\u0440\u0443\u043c\u0435\u043d\u0442 \u0432 \u043c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u0447\u0435\u0441\u043a\u043e\u043c \u0430\u043d\u0430\u043b\u0438\u0437\u0435 \u0434\u043b\u044f \u043e\u0446\u0435\u043d\u043a\u0438 \u0441\u043b\u043e\u0436\u043d\u044b\u0445 \u043f\u0440\u0435\u0434\u0435\u043b\u043e\u0432 \u0441 \u0438\u0441\u043f\u043e\u043b\u044c\u0437\u043e\u0432\u0430\u043d\u0438\u0435\u043c \u0431\u043e\u043b\u0435\u0435 \u043f\u0440\u043e\u0441\u0442\u044b\u0445 \u0444\u0443\u043d\u043a\u0446\u0438\u0439, \u043a\u043e\u0442\u043e\u0440\u044b\u0435 \u043e\u0433\u0440\u0430\u043d\u0438\u0447\u0438\u0432\u0430\u044e\u0442 \u0441\u0432\u0435\u0440\u0445\u0443 \u0438 \u0441\u043d\u0438\u0437\u0443. \u041f\u0440\u0435\u0434\u043b\u0430\u0433\u0430\u0435\u0442\u0441\u044f \u0433\u0440\u0430\u0444\u0438\u0447\u0435\u0441\u043a\u043e\u0435 \u043e\u0431\u044a\u044f\u0441\u043d\u0435\u043d\u0438\u0435 \u0438 \u0444\u043e\u0440\u043c\u0430\u043b\u044c\u043d\u043e\u0435 \u0434\u043e\u043a\u0430\u0437\u0430\u0442\u0435\u043b\u044c\u0441\u0442\u0432\u043e, \u0437\u0430 \u043a\u043e\u0442\u043e\u0440\u044b\u043c\u0438 \u0441\u043b\u0435\u0434\u0443\u044e\u0442 \u043f\u0440\u0430\u043a\u0442\u0438\u0447\u0435\u0441\u043a\u0438\u0435 \u043f\u0440\u0438\u043c\u0435\u0440\u044b. \u0426\u0435\u043b\u044c \u0441\u043e\u0441\u0442\u043e\u0438\u0442 \u0432 \u0442\u043e\u043c, \u0447\u0442\u043e\u0431\u044b \u0441\u0442\u0443\u0434\u0435\u043d\u0442\u044b \u043f\u043e\u043d\u044f\u043b\u0438, \u043a\u0430\u043a \u043f\u0440\u0438\u043c\u0435\u043d\u044f\u0442\u044c \u044d\u0442\u0443 \u0442\u0435\u043e\u0440\u0435\u043c\u0443 \u0434\u043b\u044f \u0431\u043e\u043b\u0435\u0435 \u044d\u0444\u0444\u0435\u043a\u0442\u0438\u0432\u043d\u043e\u0433\u043e \u0432\u044b\u0447\u0438\u0441\u043b\u0435\u043d\u0438\u044f \u043f\u0440\u0435\u0434\u0435\u043b\u043e\u0432.<\/em><\/p>\n<p><strong>\u0426\u0435\u043b\u0438 \u043e\u0431\u0443\u0447\u0435\u043d\u0438\u044f:<\/strong><br \/>\n\u041f\u043e\u0441\u043b\u0435 \u0437\u0430\u0432\u0435\u0440\u0448\u0435\u043d\u0438\u044f \u044d\u0442\u043e\u0433\u043e \u0443\u0440\u043e\u043a\u0430 \u0441\u0442\u0443\u0434\u0435\u043d\u0442 \u0441\u043c\u043e\u0436\u0435\u0442<\/p>\n<ul style=\"text-align:left;\">\n<li><strong>\u041f\u043e\u043d\u044f\u0442\u044c<\/strong> \u043f\u043e\u043b\u0435\u0437\u043d\u043e\u0441\u0442\u044c \u0422\u0435\u043e\u0440\u0435\u043c\u044b \u0421\u044d\u043d\u0434\u0432\u0438\u0447\u0430 \u0432 \u0432\u044b\u0447\u0438\u0441\u043b\u0435\u043d\u0438\u0438 \u043f\u0440\u0435\u0434\u0435\u043b\u043e\u0432.<\/li>\n<li><strong>\u041e\u043f\u0440\u0435\u0434\u0435\u043b\u0438\u0442\u044c<\/strong> \u0444\u0443\u043d\u043a\u0446\u0438\u0438, \u043a\u043e\u0442\u043e\u0440\u044b\u0435 \u043c\u043e\u0433\u0443\u0442 \u043e\u0433\u0440\u0430\u043d\u0438\u0447\u0438\u0432\u0430\u0442\u044c \u0446\u0435\u043b\u0435\u0432\u0443\u044e \u0444\u0443\u043d\u043a\u0446\u0438\u044e \u0434\u043b\u044f \u043f\u0440\u0438\u043c\u0435\u043d\u0435\u043d\u0438\u044f \u0442\u0435\u043e\u0440\u0435\u043c\u044b.<\/li>\n<li><strong>\u041f\u0440\u0438\u043c\u0435\u043d\u044f\u0442\u044c<\/strong> \u0422\u0435\u043e\u0440\u0435\u043c\u0443 \u0421\u044d\u043d\u0434\u0432\u0438\u0447\u0430 \u0434\u043b\u044f \u0432\u044b\u0447\u0438\u0441\u043b\u0435\u043d\u0438\u044f \u0441\u043b\u043e\u0436\u043d\u044b\u0445 \u043f\u0440\u0435\u0434\u0435\u043b\u043e\u0432.<\/li>\n<li><strong>\u0412\u0438\u0437\u0443\u0430\u043b\u0438\u0437\u0438\u0440\u043e\u0432\u0430\u0442\u044c<\/strong> \u043a\u043e\u043d\u0446\u0435\u043f\u0446\u0438\u044e \u0422\u0435\u043e\u0440\u0435\u043c\u044b \u0421\u044d\u043d\u0434\u0432\u0438\u0447\u0430 \u0433\u0440\u0430\u0444\u0438\u0447\u0435\u0441\u043a\u0438.<\/li>\n<li><strong>\u0414\u043e\u043a\u0430\u0437\u0430\u0442\u044c<\/strong> \u0422\u0435\u043e\u0440\u0435\u043c\u0443 \u0421\u044d\u043d\u0434\u0432\u0438\u0447\u0430 \u0444\u043e\u0440\u043c\u0430\u043b\u044c\u043d\u043e.<\/li>\n<\/ul>\n<p><strong><u>\u0421\u043e\u0434\u0435\u0440\u0436\u0430\u043d\u0438\u0435:<\/u><\/strong><br \/>\n<a href=\"#1\">\u0412\u0432\u0435\u0434\u0435\u043d\u0438\u0435<\/a><br \/>\n<a href=\"#2\">\u0413\u0440\u0430\u0444\u0438\u0447\u0435\u0441\u043a\u0430\u044f \u0438\u0434\u0435\u044f \u0422\u0435\u043e\u0440\u0435\u043c\u044b \u0421\u044d\u043d\u0434\u0432\u0438\u0447\u0430<\/a><br \/>\n<a href=\"#3\">\u0414\u043e\u043a\u0430\u0437\u0430\u0442\u0435\u043b\u044c\u0441\u0442\u0432\u043e \u0422\u0435\u043e\u0440\u0435\u043c\u044b \u0421\u044d\u043d\u0434\u0432\u0438\u0447\u0430<\/a><br \/>\n<a href=\"#4\">\u041f\u0440\u0438\u043c\u0435\u0440\u044b<\/a><\/p>\n<p><iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/24G_qlEwL9M\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen=\"allowfullscreen\"><\/iframe><br \/>\n<\/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; color: #000000;\"><a href=\"https:\/\/www.youtube.com\/watch?v=24G_qlEwL9M&amp;t=158s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">\u041f\u043e\u043b\u044c\u0437\u0430 \u0422\u0435\u043e\u0440\u0435\u043c\u044b \u0421\u044d\u043d\u0434\u0432\u0438\u0447\u0430 \u0437\u0430\u043a\u043b\u044e\u0447\u0430\u0435\u0442\u0441\u044f \u0432 \u0435\u0435 \u043f\u0440\u043e\u0441\u0442\u043e\u0442\u0435 \u043f\u0440\u0438 \u0432\u044b\u0447\u0438\u0441\u043b\u0435\u043d\u0438\u0438 \u043d\u0435\u043a\u043e\u0442\u043e\u0440\u044b\u0445 \u0441\u043b\u043e\u0436\u043d\u044b\u0445 \u043f\u0440\u0435\u0434\u0435\u043b\u043e\u0432<\/span><\/strong><\/a> \u0447\u0435\u0440\u0435\u0437 \u0431\u043e\u043b\u0435\u0435 \u043f\u0440\u043e\u0441\u0442\u044b\u0435 \u0444\u0443\u043d\u043a\u0446\u0438\u0438. \u041d\u0430\u0437\u0432\u0430\u043d\u0438\u0435 \u043f\u0440\u043e\u0438\u0441\u0445\u043e\u0434\u0438\u0442 \u043e\u0442 \u0442\u043e\u0433\u043e, \u0447\u0442\u043e \u0432\u043c\u0435\u0441\u0442\u043e \u043f\u0440\u044f\u043c\u043e\u0433\u043e \u0432\u044b\u0447\u0438\u0441\u043b\u0435\u043d\u0438\u044f \u043f\u0440\u0435\u0434\u0435\u043b\u0430 \u0444\u0443\u043d\u043a\u0446\u0438\u0438 \u043f\u0440\u0438 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x\\to x_0<\/span><\/span>, \u0438\u0441\u043f\u043e\u043b\u044c\u0437\u0443\u0435\u0442\u0441\u044f \u043f\u0430\u0440\u0430 \u0434\u0440\u0443\u0433\u0438\u0445 \u0444\u0443\u043d\u043a\u0446\u0438\u0439, \u043e\u0434\u043d\u0430 \u0438\u0437 \u043a\u043e\u0442\u043e\u0440\u044b\u0445 \u043e\u0433\u0440\u0430\u043d\u0438\u0447\u0438\u0432\u0430\u0435\u0442 \u0441\u0432\u0435\u0440\u0445\u0443, \u0430 \u0434\u0440\u0443\u0433\u0430\u044f \u0441\u043d\u0438\u0437\u0443, \u0438 \u0438\u0445 \u043f\u0440\u0435\u0434\u0435\u043b \u043f\u0440\u0438 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x_0<\/span><\/span> \u0441\u043e\u0432\u043f\u0430\u0434\u0430\u0435\u0442 \u0438 \u043b\u0435\u0433\u043a\u043e \u0432\u044b\u0447\u0438\u0441\u043b\u044f\u0435\u0442\u0441\u044f. \u041f\u043e\u0441\u043a\u043e\u043b\u044c\u043a\u0443 \u0438\u0441\u0445\u043e\u0434\u043d\u0430\u044f \u0444\u0443\u043d\u043a\u0446\u0438\u044f \u0432\u0441\u0435\u0433\u0434\u0430 \u043d\u0430\u0445\u043e\u0434\u0438\u0442\u0441\u044f \u043c\u0435\u0436\u0434\u0443 \u044d\u0442\u0438\u043c\u0438 \u0434\u0432\u0443\u043c\u044f, \u043e\u043d\u0430 \u043f\u043e\u0434\u043e\u0431\u043d\u0430 \u00ab\u0441\u044b\u0440\u0443 \u043c\u0435\u0436\u0434\u0443 \u0434\u0432\u0443\u043c\u044f \u043a\u0443\u0441\u043e\u0447\u043a\u0430\u043c\u0438 \u0445\u043b\u0435\u0431\u0430\u00bb.<\/p>\n<p><a name=\"2\"><\/a><\/p>\n<h2>\u0413\u0440\u0430\u0444\u0438\u0447\u0435\u0441\u043a\u0430\u044f \u0438\u0434\u0435\u044f \u0422\u0435\u043e\u0440\u0435\u043c\u044b \u0421\u044d\u043d\u0434\u0432\u0438\u0447\u0430<\/h2>\n<p style=\"text-align: justify; color: #000000;\"><a href=\"https:\/\/www.youtube.com\/watch?v=24G_qlEwL9M&amp;t=206s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">\u0418\u0434\u0435\u044f, \u0437\u0430\u043a\u043b\u044e\u0447\u0435\u043d\u043d\u0430\u044f \u0432 \u0442\u0435\u043e\u0440\u0435\u043c\u0435, \u043d\u0430 \u0441\u0430\u043c\u043e\u043c \u0434\u0435\u043b\u0435 \u0434\u043e\u0441\u0442\u0430\u0442\u043e\u0447\u043d\u043e \u043f\u0440\u043e\u0441\u0442\u0430.<\/span> <\/strong><\/a>\u041f\u0440\u0435\u0434\u043f\u043e\u043b\u043e\u0436\u0438\u043c, \u0447\u0442\u043e \u043c\u044b \u0445\u043e\u0442\u0438\u043c \u0432\u044b\u0447\u0438\u0441\u043b\u0438\u0442\u044c \u0441\u043b\u043e\u0436\u043d\u044b\u0439 \u043f\u0440\u0435\u0434\u0435\u043b<\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle\\lim_{x \\to x_0}f(x)<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">\u041e\u0431\u044b\u0447\u043d\u043e \u043c\u044b \u0438\u0441\u043f\u043e\u043b\u044c\u0437\u0443\u0435\u043c \u0432\u0441\u0435 \u043d\u0430\u0448\u0438 \u0437\u043d\u0430\u043d\u0438\u044f \u043e \u0444\u0443\u043d\u043a\u0446\u0438\u044f\u0445, \u0447\u0442\u043e\u0431\u044b \u043f\u043e\u043f\u044b\u0442\u0430\u0442\u044c\u0441\u044f <strong>\u0443\u043f\u0440\u043e\u0441\u0442\u0438\u0442\u044c \u0437\u0430\u0434\u0430\u0447\u0443 \u0434\u043e \u0442\u043e\u0433\u043e \u043c\u043e\u043c\u0435\u043d\u0442\u0430, \u043a\u043e\u0433\u0434\u0430 \u0441\u043c\u043e\u0436\u0435\u043c \u0435\u0435 \u0440\u0435\u0448\u0438\u0442\u044c<\/strong>. \u041e\u0434\u043d\u0430\u043a\u043e \u0432 \u043d\u0435\u043a\u043e\u0442\u043e\u0440\u044b\u0445 \u0441\u043b\u0443\u0447\u0430\u044f\u0445 \u0434\u0440\u0443\u0433\u043e\u0439 \u043f\u043e\u0434\u0445\u043e\u0434 \u043e\u043a\u0430\u0437\u044b\u0432\u0430\u0435\u0442\u0441\u044f \u0431\u043e\u043b\u0435\u0435 \u044d\u0444\u0444\u0435\u043a\u0442\u0438\u0432\u043d\u044b\u043c. \u041f\u0440\u0435\u0434\u043f\u043e\u043b\u043e\u0436\u0438\u043c, \u0447\u0442\u043e \u0443 \u043d\u0430\u0441 \u0435\u0441\u0442\u044c \u0437\u0430\u043a\u0440\u044b\u0442\u044b\u0439 \u0438\u043d\u0442\u0435\u0440\u0432\u0430\u043b <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">I<\/span><\/span>, \u0433\u0434\u0435 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x_0 \\in I<\/span><\/span>, \u0438 \u0441\u0443\u0449\u0435\u0441\u0442\u0432\u0443\u044e\u0442 \u0435\u0449\u0435 \u0434\u0432\u0435 \u0444\u0443\u043d\u043a\u0446\u0438\u0438 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">m(x)<\/span><\/span> \u0438 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">M(x)<\/span><\/span>, \u043a\u043e\u0442\u043e\u0440\u044b\u0435 \u0443\u0434\u043e\u0432\u043b\u0435\u0442\u0432\u043e\u0440\u044f\u044e\u0442 \u0441\u043b\u0435\u0434\u0443\u044e\u0449\u0435\u043c\u0443 \u0443\u0441\u043b\u043e\u0432\u0438\u044e<\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(\\forall x\\in I)(m(x)\\leq f(x) \\leq M(x) )<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">\u0418 \u0442\u0430\u043a\u0436\u0435<\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to x_0} m(x) = \\lim_{x\\to x_0} M(x) = L<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">\u0422\u043e\u0433\u0434\u0430 \u0432\u044b\u043f\u043e\u043b\u043d\u044f\u0435\u0442\u0441\u044f<\/p>\n<p style=\"text-align: center; color: #000000;\"><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: justify; color: #000000;\">\u042d\u0442\u043e \u043c\u043e\u0436\u043d\u043e \u0443\u0432\u0438\u0434\u0435\u0442\u044c \u043d\u0430 \u0441\u043b\u0435\u0434\u0443\u044e\u0449\u0435\u043c \u0438\u0437\u043e\u0431\u0440\u0430\u0436\u0435\u043d\u0438\u0438.<\/p>\n<p><center><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/1.bp.blogspot.com\/-gjBfVdaLj-k\/YGDXVUQBqDI\/AAAAAAAAEvg\/d2sNJdweVaoB64O5e2qfBxjZGIyIyGOxgCLcBGAsYHQ\/s0\/teorema%2Bdel%2Bsandwich.PNG\" alt=\"\u0422\u0435\u043e\u0440\u0435\u043c\u0430 \u0421\u044d\u043d\u0434\u0432\u0438\u0447\u0430\" class=\"alignnone size-full lazyload\" width=\"513\" height=\"407\" \/><noscript><img decoding=\"async\" src=\"https:\/\/1.bp.blogspot.com\/-gjBfVdaLj-k\/YGDXVUQBqDI\/AAAAAAAAEvg\/d2sNJdweVaoB64O5e2qfBxjZGIyIyGOxgCLcBGAsYHQ\/s0\/teorema%2Bdel%2Bsandwich.PNG\" alt=\"\u0422\u0435\u043e\u0440\u0435\u043c\u0430 \u0421\u044d\u043d\u0434\u0432\u0438\u0447\u0430\" class=\"alignnone size-full lazyload\" width=\"513\" height=\"407\" \/><\/noscript><\/center><\/p>\n<p><a name=\"3\"><\/a><\/p>\n<h2>\u0414\u043e\u043a\u0430\u0437\u0430\u0442\u0435\u043b\u044c\u0441\u0442\u0432\u043e \u0422\u0435\u043e\u0440\u0435\u043c\u044b \u0421\u044d\u043d\u0434\u0432\u0438\u0447\u0430<\/h2>\n<p style=\"text-align: justify; color: #000000;\"><a href=\"https:\/\/www.youtube.com\/watch?v=24G_qlEwL9M&amp;t=404s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">\u0414\u043b\u044f \u0434\u043e\u043a\u0430\u0437\u0430\u0442\u0435\u043b\u044c\u0441\u0442\u0432\u0430 \u0422\u0435\u043e\u0440\u0435\u043c\u044b \u0421\u044d\u043d\u0434\u0432\u0438\u0447\u0430<\/span><\/strong><\/a>, \u043c\u044b \u0431\u0443\u0434\u0435\u043c \u0441\u043b\u0435\u0434\u043e\u0432\u0430\u0442\u044c \u0441\u043b\u0435\u0434\u0443\u044e\u0449\u0435\u043c\u0443 \u0440\u0430\u0441\u0441\u0443\u0436\u0434\u0435\u043d\u0438\u044e:<\/p>\n<table style=\"text-align: justify; color: #000000;\">\n<tbody>\n<tr>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(1)<\/span><\/span><\/td>\n<td><span style=\"background-color: #90ff90;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x_0\\in I<\/span><\/span><\/span>; <strong>\u041f\u043e\u0441\u044b\u043b\u043a\u0430<\/strong><\/td>\n<\/tr>\n<tr>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(2)<\/span><\/span><\/td>\n<td><span style=\"background-color: #90ff90;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to x_0} m(x) = L<\/span><\/span><\/span> ; <strong>\u041f\u043e\u0441\u044b\u043b\u043a\u0430<\/strong><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(\\forall \\epsilon \\gt 0)(\\exists \\delta_1 \\gt 0) (|x-x_0|\\lt \\delta_1 \\rightarrow |m(x) -L| \\lt \\epsilon )<\/span><\/span><\/td>\n<\/tr>\n<tr>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(3)<\/span><\/span><\/td>\n<td><span style=\"background-color: #90ff90;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to x_0} M(x) = L<\/span><\/span><\/span> ; <strong>\u041f\u043e\u0441\u044b\u043b\u043a\u0430<\/strong><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(\\forall \\epsilon \\gt 0)(\\exists \\delta_2 \\gt 0) (|x-x_0|\\lt \\delta_2 \\rightarrow |M(x) -L| \\lt \\epsilon )<\/span><\/span><\/td>\n<\/tr>\n<tr>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(4)<\/span><\/span><\/td>\n<td><span style=\"background-color: #90ff90;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(\\forall x \\in I)(m(x) \\leq f(x) \\leq M(x) )<\/span><\/span><\/span>; <strong>\u041f\u043e\u0441\u044b\u043b\u043a\u0430<\/strong><\/td>\n<\/tr>\n<tr>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(5)<\/span><\/span><\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(\\forall x \\in I)(m(x) - L \\leq f(x) - L \\leq M(x) - L )<\/span><\/span>; \u0418\u0437 (4)<\/td>\n<\/tr>\n<tr>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(6)<\/span><\/span><\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(|m(x) -L|\\lt \\epsilon) \\rightarrow (-\\epsilon \\lt m(x) - L \\lt \\epsilon)<\/span><\/span><\/td>\n<\/tr>\n<tr>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(7)<\/span><\/span><\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(|M(x) -L|\\lt \\epsilon ) \\rightarrow (-\\epsilon \\lt M(x) - L \\lt \\epsilon) <\/span><\/span><\/td>\n<\/tr>\n<tr>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(8)<\/span><\/span><\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(\\forall \\epsilon \\gt 0)(\\exists \\delta \\gt 0) (|x-x_0|\\lt \\delta=\\min\\{\\delta_1,\\delta_2\\} \\rightarrow ( |M(x) -L| \\lt \\epsilon \\wedge |m(x) -L| \\lt \\epsilon ) )<\/span><\/span>; \u0418\u0437 (2,3)<\/td>\n<\/tr>\n<tr>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(9)<\/span><\/span><\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(\\forall \\epsilon \\gt 0)(\\exists \\delta \\gt 0) (|x-x_0|\\lt \\delta=\\min\\{\\delta_1,\\delta_2\\} \\rightarrow ( - \\epsilon \\lt f(x) - L \\lt \\epsilon ) )<\/span><\/span>; \u0418\u0437 (1,5,6,7,8)<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(\\forall \\epsilon \\gt 0)(\\exists \\delta \\gt 0) (|x-x_0|\\lt \\delta=\\min\\{\\delta_1,\\delta_2\\} \\rightarrow |f(x) - L| \\lt \\epsilon ) )<\/span><\/span><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to x_0}f(x) = L\\;\\blacksquare<\/span><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><a name=\"4\"><\/a><\/p>\n<h2>\u041f\u0440\u0438\u043c\u0435\u0440\u044b<\/h2>\n<p style=\"text-align: justify; color: #000000;\">\u0418\u0441\u043f\u043e\u043b\u044c\u0437\u0443\u044f \u0422\u0435\u043e\u0440\u0435\u043c\u0443 \u0421\u044d\u043d\u0434\u0432\u0438\u0447\u0430, \u043c\u044b \u043c\u043e\u0436\u0435\u043c \u0432\u044b\u0447\u0438\u0441\u043b\u044f\u0442\u044c \u043f\u0440\u0435\u0434\u0435\u043b\u044b \u0444\u0443\u043d\u043a\u0446\u0438\u0439, \u0434\u0430\u0436\u0435 \u0435\u0441\u043b\u0438 \u0443 \u043d\u0430\u0441 \u043d\u0435\u0442 \u0438\u0445 \u044f\u0432\u043d\u043e\u0433\u043e \u0430\u043b\u0433\u0435\u0431\u0440\u0430\u0438\u0447\u0435\u0441\u043a\u043e\u0433\u043e \u0432\u044b\u0440\u0430\u0436\u0435\u043d\u0438\u044f. \u041f\u0440\u0438\u043c\u0435\u0440\u043e\u043c \u044d\u0442\u043e\u0433\u043e \u044f\u0432\u043b\u044f\u0435\u0442\u0441\u044f \u0441\u043b\u0435\u0434\u0443\u044e\u0449\u0430\u044f \u0441\u0438\u0442\u0443\u0430\u0446\u0438\u044f:<\/p>\n<p style=\"text-align: justify; color: #000000;\">\u041f\u0440\u0438\u043c\u0435\u0440\u043e\u043c \u044d\u0442\u043e\u0433\u043e \u044f\u0432\u043b\u044f\u0435\u0442\u0441\u044f \u0441\u043b\u0435\u0434\u0443\u044e\u0449\u0430\u044f \u0441\u0438\u0442\u0443\u0430\u0446\u0438\u044f:<\/p>\n<ul style=\"text-align: justify; color: #000000;\">\n<li>\u0415\u0441\u043b\u0438 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\sqrt{5-2x^2}\\leq f(x) \\leq \\sqrt{5-x^2}<\/span><\/span>, \u043f\u0440\u0438 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">-1\\leq x\\leq 1<\/span><\/span>. \u041a\u0430\u043a\u043e\u0432\u043e \u0437\u043d\u0430\u0447\u0435\u043d\u0438\u0435 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to 0}f(x)<\/span><\/span>? <a href=\"https:\/\/www.youtube.com\/watch?v=24G_qlEwL9M&amp;t=1082s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">[\u0420\u0415\u0428\u0415\u041d\u0418\u0415]<\/span><\/strong><\/a><\/li>\n<\/ul>\n<p style=\"text-align: justify; color: #000000;\">\u0415\u0449\u0435 \u043e\u0434\u043d\u043e \u043f\u0440\u0430\u043a\u0442\u0438\u0447\u0435\u0441\u043a\u043e\u0435 \u043f\u0440\u0438\u043c\u0435\u043d\u0435\u043d\u0438\u0435 \u0422\u0435\u043e\u0440\u0435\u043c\u044b \u0421\u044d\u043d\u0434\u0432\u0438\u0447\u0430 \u0437\u0430\u043a\u043b\u044e\u0447\u0430\u0435\u0442\u0441\u044f \u0432 \u0442\u043e\u043c, \u0447\u0442\u043e \u043f\u0440\u0435\u0434\u0435\u043b \u0441\u0430\u043c \u043f\u043e \u0441\u0435\u0431\u0435 \u043d\u0435 \u043e\u0447\u0435\u0432\u0438\u0434\u0435\u043d \u043f\u043e \u0441\u0440\u0430\u0432\u043d\u0435\u043d\u0438\u044e \u0441 \u0434\u0440\u0443\u0433\u0438\u043c\u0438 \u0431\u043e\u043b\u0435\u0435 \u043f\u0440\u043e\u0441\u0442\u044b\u043c\u0438 \u043f\u0440\u0435\u0434\u0435\u043b\u0430\u043c\u0438, \u043a\u043e\u0442\u043e\u0440\u044b\u0435 \u043e\u0433\u0440\u0430\u043d\u0438\u0447\u0438\u0432\u0430\u044e\u0442 \u0435\u0433\u043e \u0441\u0432\u0435\u0440\u0445\u0443 \u0438 \u0441\u043d\u0438\u0437\u0443, \u043a\u0430\u043a \u044d\u0442\u043e \u043f\u0440\u043e\u0438\u0441\u0445\u043e\u0434\u0438\u0442 \u043f\u0440\u0438 \u0432\u044b\u0447\u0438\u0441\u043b\u0435\u043d\u0438\u0438 \u0441\u043b\u0435\u0434\u0443\u044e\u0449\u0435\u0433\u043e \u0441\u043b\u0443\u0447\u0430\u044f:<\/p>\n<ul style=\"text-align: justify; color: #000000;\">\n<li>\u0412\u044b\u0447\u0438\u0441\u043b\u0438\u0442\u044c: <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to 0}\\dfrac{\\sin(x)}{x}<\/span><\/span> <a href=\"https:\/\/www.youtube.com\/watch?v=24G_qlEwL9M&amp;t=1157s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">[\u0420\u0415\u0428\u0415\u041d\u0418\u0415]<\/span><\/strong><\/a><\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>\u0422\u0435\u043e\u0440\u0435\u043c\u0430 \u0421\u044d\u043d\u0434\u0432\u0438\u0447\u0430 \u0434\u043b\u044f \u0432\u044b\u0447\u0438\u0441\u043b\u0435\u043d\u0438\u044f \u043f\u0440\u0435\u0434\u0435\u043b\u043e\u0432 \u0420\u0435\u0437\u044e\u043c\u0435: \u042d\u0442\u043e\u0442 \u0443\u0440\u043e\u043a \u043f\u0440\u0435\u0434\u0441\u0442\u0430\u0432\u043b\u044f\u0435\u0442 \u0422\u0435\u043e\u0440\u0435\u043c\u0443 \u0421\u044d\u043d\u0434\u0432\u0438\u0447\u0430, \u043a\u043b\u044e\u0447\u0435\u0432\u043e\u0439 \u0438\u043d\u0441\u0442\u0440\u0443\u043c\u0435\u043d\u0442 \u0432 \u043c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u0447\u0435\u0441\u043a\u043e\u043c \u0430\u043d\u0430\u043b\u0438\u0437\u0435 \u0434\u043b\u044f \u043e\u0446\u0435\u043d\u043a\u0438 \u0441\u043b\u043e\u0436\u043d\u044b\u0445 \u043f\u0440\u0435\u0434\u0435\u043b\u043e\u0432 \u0441 \u0438\u0441\u043f\u043e\u043b\u044c\u0437\u043e\u0432\u0430\u043d\u0438\u0435\u043c \u0431\u043e\u043b\u0435\u0435 \u043f\u0440\u043e\u0441\u0442\u044b\u0445 \u0444\u0443\u043d\u043a\u0446\u0438\u0439, \u043a\u043e\u0442\u043e\u0440\u044b\u0435 \u043e\u0433\u0440\u0430\u043d\u0438\u0447\u0438\u0432\u0430\u044e\u0442 \u0441\u0432\u0435\u0440\u0445\u0443 \u0438 \u0441\u043d\u0438\u0437\u0443. \u041f\u0440\u0435\u0434\u043b\u0430\u0433\u0430\u0435\u0442\u0441\u044f \u0433\u0440\u0430\u0444\u0438\u0447\u0435\u0441\u043a\u043e\u0435 \u043e\u0431\u044a\u044f\u0441\u043d\u0435\u043d\u0438\u0435 \u0438 \u0444\u043e\u0440\u043c\u0430\u043b\u044c\u043d\u043e\u0435 \u0434\u043e\u043a\u0430\u0437\u0430\u0442\u0435\u043b\u044c\u0441\u0442\u0432\u043e, \u0437\u0430 \u043a\u043e\u0442\u043e\u0440\u044b\u043c\u0438 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