{"id":27827,"date":"2024-08-11T13:00:30","date_gmt":"2024-08-11T13:00:30","guid":{"rendered":"http:\/\/toposuranos.com\/material\/?p=27827"},"modified":"2024-09-12T19:47:08","modified_gmt":"2024-09-12T19:47:08","slug":"%d8%ad%d8%af-%d8%a7%d9%84%d8%af%d9%88%d8%a7%d9%84-%d9%84%d9%85%d8%aa%d8%ba%d9%8a%d8%b1-%d8%ad%d9%82%d9%8a%d9%82%d9%8a-%d9%88%d8%a7%d8%ad%d8%af","status":"publish","type":"post","link":"https:\/\/toposuranos.com\/material\/ar\/%d8%ad%d8%af-%d8%a7%d9%84%d8%af%d9%88%d8%a7%d9%84-%d9%84%d9%85%d8%aa%d8%ba%d9%8a%d8%b1-%d8%ad%d9%82%d9%8a%d9%82%d9%8a-%d9%88%d8%a7%d8%ad%d8%af\/","title":{"rendered":"\u062d\u062f \u0627\u0644\u062f\u0648\u0627\u0644 \u0644\u0645\u062a\u063a\u064a\u0631 \u062d\u0642\u064a\u0642\u064a \u0648\u0627\u062d\u062f"},"content":{"rendered":"<p><center><\/p>\n<h1>\u062d\u062f\u0648\u062f \u0627\u0644\u062f\u0648\u0627\u0644 \u0630\u0627\u062a \u0627\u0644\u0645\u062a\u063a\u064a\u0631 \u0627\u0644\u062d\u0642\u064a\u0642\u064a<\/h1>\n<p><\/center><\/p>\n<p style=\"text-align:center\"><em><strong>\u0645\u0644\u062e\u0635:<\/strong><br \/>\n\u0641\u064a \u0647\u0630\u0647 \u0627\u0644\u062d\u0635\u0629\u060c \u0633\u0646\u0642\u0648\u0645 \u0628\u0645\u0631\u0627\u062c\u0639\u0629 \u0634\u0627\u0645\u0644\u0629 \u0644\u0644\u062a\u0639\u0631\u064a\u0641 \u0627\u0644\u0631\u0633\u0645\u064a \u0644\u062d\u062f\u0648\u062f \u0627\u0644\u062f\u0648\u0627\u0644 \u0630\u0627\u062a \u0627\u0644\u0645\u062a\u063a\u064a\u0631 \u0627\u0644\u062d\u0642\u064a\u0642\u064a\u060c \u0648\u0645\u0646 \u062e\u0644\u0627\u0644 \u0630\u0644\u0643 \u0633\u0646\u062b\u0628\u062a \u0627\u0644\u062e\u0635\u0627\u0626\u0635 \u0627\u0644\u0631\u0626\u064a\u0633\u064a\u0629 \u0627\u0644\u062a\u064a \u062a\u0642\u0648\u062f \u0625\u0644\u0649 \u062c\u0628\u0631 \u0627\u0644\u062d\u062f\u0648\u062f.<\/br><\/em><\/p>\n<p style=\"text-align:center\"><em><strong>\u0623\u0647\u062f\u0627\u0641 \u0627\u0644\u062a\u0639\u0644\u0645:<\/strong><br \/>\n\u0641\u064a \u0646\u0647\u0627\u064a\u0629 \u0647\u0630\u0647 \u0627\u0644\u062d\u0635\u0629\u060c \u0633\u064a\u0643\u0648\u0646 \u0627\u0644\u0637\u0627\u0644\u0628 \u0642\u0627\u062f\u0631\u064b\u0627 \u0639\u0644\u0649:<\/p>\n<ul>\n<li><strong>\u062a\u0630\u0643\u0631<\/strong> \u062a\u0639\u0631\u064a\u0641 \u062d\u062f \u0627\u0644\u062f\u0648\u0627\u0644 \u0630\u0627\u062a \u0627\u0644\u0645\u062a\u063a\u064a\u0631 \u0627\u0644\u062d\u0642\u064a\u0642\u064a.<\/li>\n<li><strong>\u0625\u062b\u0628\u0627\u062a<\/strong> \u0627\u0644\u062e\u0635\u0627\u0626\u0635 \u0627\u0644\u062a\u064a \u062a\u0642\u0648\u062f \u0625\u0644\u0649 \u062c\u0628\u0631 \u0627\u0644\u062d\u062f\u0648\u062f \u0645\u0646 \u062e\u0644\u0627\u0644 \u0627\u0633\u062a\u0646\u062a\u0627\u062c\u0627\u062a <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\epsilon-\\delta<\/span><\/span>.<\/li>\n<li><strong>\u062d\u0633\u0627\u0628<\/strong> \u062d\u062f\u0648\u062f \u0627\u0644\u062f\u0648\u0627\u0644 \u0630\u0627\u062a \u0627\u0644\u0645\u062a\u063a\u064a\u0631 \u0627\u0644\u062d\u0642\u064a\u0642\u064a \u0628\u0627\u0633\u062a\u062e\u062f\u0627\u0645 \u062c\u0628\u0631 \u0627\u0644\u062d\u062f\u0648\u062f \u0648\u062e\u0635\u0627\u0626\u0635\u0647.<\/li>\n<\/ul>\n<p><\/em><\/p>\n<p><center><br \/>\n<strong>\u0641\u0647\u0631\u0633 \u0627\u0644\u0645\u062d\u062a\u0648\u064a\u0627\u062a<\/strong><br \/>\n<a href=\"#1\"><strong>\u0645\u0642\u062f\u0645\u0629<\/strong><\/a><br \/>\n<a href=\"#2\"><strong>\u0627\u0644\u0645\u0641\u0647\u0648\u0645 \u0627\u0644\u0628\u062f\u064a\u0647\u064a \u0644\u0644\u062d\u062f \u0645\u0646 \u0645\u0646\u0638\u0648\u0631 \u0631\u0633\u0648\u0645\u064a<\/strong><\/a><br \/>\n<a href=\"#3\"><strong>\u0627\u0644\u062a\u0639\u0631\u064a\u0641 \u0627\u0644\u0631\u0633\u0645\u064a \u0644\u0644\u062d\u062f<\/strong><\/a><br \/>\n<a href=\"#4\"><strong>\u062e\u0635\u0627\u0626\u0635 \u0627\u0644\u062d\u062f\u0648\u062f<\/strong><\/a><br \/>\n<a href=\"#5\">\u0625\u0630\u0627 \u0643\u0627\u0646 \u0627\u0644\u062d\u062f \u0645\u0648\u062c\u0648\u062f\u064b\u0627\u060c \u0641\u0625\u0646\u0647 \u064a\u0643\u0648\u0646 \u0648\u062d\u064a\u062f\u064b\u0627<\/a><br \/>\n<a href=\"#6\">\u062c\u0628\u0631 \u0627\u0644\u062d\u062f\u0648\u062f<\/a><br \/>\n<a href=\"#7\">\u062d\u0633\u0627\u0628 \u0627\u0644\u062d\u062f\u0648\u062f \u0627\u0644\u0628\u0633\u064a\u0637\u0629<\/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>\u0645\u0642\u062f\u0645\u0629<\/h2>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=FEPfoAfPsFY&amp;t=240s\" rel=\"noopener\" target=\"_blank\"><strong><\u0645\u0627 \u0627\u0644\u0641\u0631\u0642 \u0628\u064a\u0646 \u062f\u0631\u0627\u0633\u0629 \u0627\u0644\u062c\u0628\u0631 \u0648\u0627\u0644\u0647\u0646\u062f\u0633\u0629 \u0641\u064a\u0645\u0627 \u064a\u062a\u0639\u0644\u0642 \u0628\u062f\u0631\u0627\u0633\u0629 \u0627\u0644\u062d\u0633\u0627\u0628 \u0627\u0644\u062a\u0641\u0627\u0636\u0644\u064a\u061f<\/strong><\/a> \u0627\u0644\u062c\u0648\u0627\u0628 \u0639\u0644\u0649 \u0647\u0630\u0627 \u0627\u0644\u0633\u0624\u0627\u0644 \u064a\u0639\u0637\u064a\u0646\u0627 \u0645\u0641\u0647\u0648\u0645 \u0627\u0644\u062d\u062f. \u0641\u064a \u0647\u0630\u0647 \u0627\u0644\u0645\u0642\u0627\u0644\u0629\u060c \u0646\u062f\u0631\u0633 \u0628\u0627\u0644\u062a\u0627\u0644\u064a \u0627\u0644\u062d\u062f \u0648\u062a\u0639\u0631\u064a\u0641\u0647.<\/p>\n<p style=\"text-align: justify;\">\u0639\u0627\u062f\u0629\u064b \u0645\u0627 \u0646\u0631\u0628\u0637 \u0643\u0644\u0645\u0629 \u00ab\u062d\u062f\u00bb \u0628\u0646\u0648\u0639 \u0645\u0646 \u0627\u0644\u062d\u062f\u0648\u062f\u060c \u0645\u062b\u0644 \u062d\u062f\u0648\u062f \u0641\u062a\u0631\u0629 \u0646\u0647\u0627\u064a\u062a\u0647\u0627 a \u0648b (\u0628\u063a\u0636 \u0627\u0644\u0646\u0638\u0631 \u0639\u0646 \u0637\u0628\u064a\u0639\u062a\u0647\u0627)<\/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;\">\u0623\u0648 \u0645\u062b\u0644 \u0627\u0644\u062d\u0627\u0636\u0631\u060c \u0627\u0644\u0630\u064a \u064a\u0645\u0643\u0646\u0646\u0627 \u0627\u0644\u0642\u0648\u0644 \u0628\u0623\u0646\u0647 \u0627\u0644\u062d\u062f \u0628\u064a\u0646 \u0627\u0644\u0645\u0627\u0636\u064a \u0648\u0627\u0644\u0645\u0633\u062a\u0642\u0628\u0644. \u0628\u0634\u0643\u0644 \u0645\u0634\u0627\u0628\u0647 \u0625\u0644\u0649 \u062d\u062f \u0645\u0627\u060c \u064a\u064f\u0642\u062f\u0645 \u0645\u0641\u0647\u0648\u0645 \u0627\u0644\u062d\u062f \u0627\u0644\u0641\u0647\u0645 \u0627\u0644\u0631\u064a\u0627\u0636\u064a \u0644\u0647\u0630\u0647 \u0627\u0644\u0641\u0643\u0631\u0629 \u0627\u0644\u0628\u062f\u064a\u0647\u064a\u0629 \u062d\u0648\u0644 \u0627\u0644\u0627\u0642\u062a\u0631\u0627\u0628 \u0627\u0644\u0644\u0627\u0646\u0647\u0627\u0626\u064a \u0645\u0646 \u0646\u0642\u0637\u0629 \u0645\u0639\u064a\u0646\u0629.<\/p>\n<p><a name=\"2\"><\/a><\/p>\n<h2>\u0627\u0644\u0645\u0641\u0647\u0648\u0645 \u0627\u0644\u0628\u062f\u064a\u0647\u064a \u0644\u0644\u062d\u062f \u0645\u0646 \u0645\u0646\u0638\u0648\u0631 \u0631\u0633\u0648\u0645\u064a<\/h2>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=FEPfoAfPsFY&amp;t=314s\" rel=\"noopener\" target=\"_blank\"><strong>\u0644\u0628\u062f\u0621 \u062a\u0635\u0648\u0631 \u0641\u0643\u0631\u0629 \u0627\u0644\u062d\u062f\u060c \u0645\u0646 \u0627\u0644\u0645\u0646\u0627\u0633\u0628 \u0627\u0644\u0628\u062f\u0621 \u0628\u0627\u0644\u062a\u0645\u062b\u064a\u0644 \u0627\u0644\u0631\u0633\u0648\u0645\u064a<\/strong><\/a> \u0644\u062f\u0627\u0644\u0629 \u0645\u0627 \u0648\u0646\u0633\u0623\u0644 \u0645\u0627\u0630\u0627 \u0633\u064a\u062d\u062f\u062b \u0644\u0640 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f(x)<\/span><\/span> \u0639\u0646\u062f\u0645\u0627 \u064a\u0642\u062a\u0631\u0628 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x<\/span><\/span> \u0645\u0646 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x_0<\/span><\/span> \u0643\u0645\u0627 \u0646\u0631\u064a\u062f.<\/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=\"\u062d\u062f \u0627\u0644\u062f\u0627\u0644\u0629\" 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=\"\u062d\u062f \u0627\u0644\u062f\u0627\u0644\u0629\" class=\"alignnone size-full lazyload\" width=\"692\" height=\"565\" \/><\/noscript><\/center><\/p>\n<p style=\"text-align: justify;\">\u0625\u0630\u0627 \u0643\u0627\u0646 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x<\/span><\/span> \u0642\u0631\u064a\u0628\u064b\u0627 \u0645\u0646 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x_0<\/span><\/span>\u060c \u0641\u0633\u064a\u0643\u0648\u0646 \u0647\u0646\u0627\u0643 \u0641\u062a\u0631\u0629 \u0645\u0641\u062a\u0648\u062d\u0629 \u0646\u0635\u0641 \u0642\u0637\u0631\u0647\u0627 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\delta<\/span><\/span> \u0648\u0645\u0631\u0643\u0632\u0647\u0627 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x_0<\/span><\/span> \u0628\u062d\u064a\u062b \u064a\u0643\u0648\u0646 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x<\/span><\/span> \u0645\u0648\u062c\u0648\u062f\u064b\u0627 \u062f\u0627\u062e\u0644\u0647\u0627. \u064a\u0645\u0643\u0646\u0646\u0627 \u062a\u0645\u062b\u064a\u0644 \u0647\u0630\u0627 \u0628\u062b\u0644\u0627\u062b \u0637\u0631\u0642 \u0645\u062e\u062a\u0644\u0641\u0629:<\/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;\">\u0623\u0648 <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>\u0641\u064a \u0633\u064a\u0627\u0642\u0646\u0627\u060c \u0647\u0630\u0647 \u062b\u0644\u0627\u062b \u0637\u0631\u0642 \u0644\u0642\u0648\u0644 \u0627\u0644\u0634\u064a\u0621 \u0646\u0641\u0633\u0647\u061b \u0631\u063a\u0645 \u0623\u0646 \u0627\u0644\u0637\u0631\u064a\u0642\u0629 \u0627\u0644\u0623\u062e\u064a\u0631\u0629\u060c \u0627\u0644\u062a\u064a \u062a\u064f\u0642\u0631\u0623 \u0639\u0644\u0649 \u0623\u0646\u0647\u0627 \u00ab<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x<\/span><\/span> \u0645\u0648\u062c\u0648\u062f \u062f\u0627\u062e\u0644 \u0627\u0644\u0643\u0631\u0629 \u0627\u0644\u0645\u0641\u062a\u0648\u062d\u0629 \u0630\u0627\u062a \u0627\u0644\u0645\u0631\u0643\u0632 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x_0<\/span><\/span> \u0648\u0646\u0635\u0641 \u0627\u0644\u0642\u0637\u0631 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\delta<\/span><\/span>\u060c \u0633\u062a\u0643\u0648\u0646 \u0623\u0643\u062b\u0631 \u0645\u0644\u0627\u0621\u0645\u0629 \u0644\u062f\u0648\u0631\u0629 \u0641\u064a \u0627\u0644\u0637\u0648\u0628\u0648\u0644\u0648\u062c\u064a\u0627\u060c \u062d\u064a\u062b \u064a\u062a\u0645 \u062a\u0639\u0645\u0642 \u0623\u0643\u062b\u0631 \u0641\u064a \u00ab\u0645\u0648\u0636\u0648\u0639 \u0627\u0644\u0642\u0631\u0628\u00bb.<\/em><\/p>\n<p style=\"text-align: justify;\">\u0625\u0630\u0627 \u062d\u062f\u062b \u0647\u0630\u0627\u060c \u0641\u0633\u0646\u0644\u0627\u062d\u0638 \u0623\u0646\u0647 \u0633\u064a\u0643\u0648\u0646 \u0647\u0646\u0627\u0643 \u0641\u062a\u0631\u0629 \u0645\u0641\u062a\u0648\u062d\u0629 \u0623\u062e\u0631\u0649 \u0645\u0631\u0643\u0632\u0647\u0627 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">l<\/span><\/span> \u0648\u0646\u0635\u0641 \u0642\u0637\u0631\u0647\u0627 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\epsilon<\/span><\/span> \u0628\u062d\u064a\u062b \u062a\u0643\u0648\u0646 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f(x)<\/span><\/span> \u0645\u0648\u062c\u0648\u062f\u0629 \u062f\u0627\u062e\u0644\u0647\u0627\u060c \u0623\u064a: <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=\"\u062d\u062f \u0627\u0644\u062f\u0627\u0644\u0629\" 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=\"\u062d\u062f \u0627\u0644\u062f\u0627\u0644\u0629\" class=\"alignnone size-full lazyload\" width=\"625\" height=\"549\" \/><\/noscript><\/center><\/p>\n<p style=\"text-align: justify;\">\u0645\u0646 \u0647\u0646\u0627 \u062a\u0646\u0628\u062b\u0642 \u0627\u0644\u0641\u0643\u0631\u0629 \u0627\u0644\u0623\u0633\u0627\u0633\u064a\u0629 \u0644\u0645\u0641\u0647\u0648\u0645 \u0627\u0644\u062d\u062f \u0627\u0644\u0631\u064a\u0627\u0636\u064a\u060c \u0627\u0644\u0630\u064a \u064a\u0648\u062c\u062f \u0639\u0646\u062f\u0645\u0627: \u0625\u0630\u0627 \u0643\u0627\u0646 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0 \\lt|x-x_0|\\lt \\delta<\/span><\/span>\u060c \u0641\u0625\u0646 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">|f(x)-l|\\lt \\epsilon<\/span><\/span>\u061b \u0647\u0630\u0627 \u0627\u0644\u0645\u0642\u062f\u0627\u0631 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">l<\/span><\/span> \u0633\u064a\u0643\u0648\u0646 \u062d\u062f \u0627\u0644\u062f\u0627\u0644\u0629 \u0639\u0646\u062f\u0645\u0627 \u064a\u0642\u062a\u0631\u0628 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x<\/span><\/span> \u0645\u0646 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x_0<\/span><\/span> \u0628\u0642\u062f\u0631 \u0645\u0627 \u0646\u0631\u064a\u062f.<\/p>\n<p><a name=\"3\"><\/a><\/p>\n<h2>\u0627\u0644\u062a\u0639\u0631\u064a\u0641 \u0627\u0644\u0631\u0633\u0645\u064a \u0644\u0644\u062d\u062f<\/h2>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=FEPfoAfPsFY&amp;t=689s\" rel=\"noopener\" target=\"_blank\"><strong>\u0645\u0646 \u0627\u0644\u0645\u0641\u0647\u0648\u0645 \u0627\u0644\u0628\u062f\u064a\u0647\u064a \u0648\u0627\u0644\u0631\u0633\u0648\u0645\u064a \u0627\u0644\u0630\u064a \u062a\u0645 \u062a\u0642\u062f\u064a\u0645\u0647 \u0644\u0644\u062a\u0648\u060c \u064a\u0645\u0643\u0646\u0646\u0627 \u0627\u0644\u0628\u062f\u0621 \u0641\u064a \u0627\u0633\u062a\u0646\u0628\u0627\u0637 \u0627\u0644\u062a\u0639\u0631\u064a\u0641 \u0627\u0644\u0631\u0633\u0645\u064a \u0644\u0644\u062d\u062f.<\/strong> <\/a> \u0646\u0642\u0648\u0644 \u0623\u0646 \u0627\u0644\u062d\u062f \u0645\u0648\u062c\u0648\u062f \u0639\u0646\u062f\u0645\u0627\u060c \u0628\u063a\u0636 \u0627\u0644\u0646\u0638\u0631 \u0639\u0646 \u0645\u0646 \u064a\u0643\u0648\u0646 \u0647\u0630\u0627 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\epsilon<\/span><\/span> (\u0623\u064a\u060c \u0627\u0644\u0645\u0633\u0627\u0641\u0629 \u0628\u064a\u0646 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f(x)<\/span><\/span> \u0648 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">l<\/span><\/span>)\u060c \u0633\u064a\u0643\u0648\u0646 \u062f\u0627\u0626\u0645\u064b\u0627 \u0647\u0646\u0627\u0643 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\delta<\/span><\/span> \u0628\u062d\u064a\u062b \u0625\u0630\u0627 \u0643\u0627\u0646 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0 \\lt|x-x_0|\\lt \\delta<\/span><\/span>\u060c \u0641\u0625\u0646 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">|f(x) - l|\\lt \\epsilon.<\/span><\/span> \u0647\u0630\u0647 \u0627\u0644\u0641\u0643\u0631\u0629\u060c \u0627\u0644\u062a\u064a \u064a\u0635\u0639\u0628 \u0627\u0633\u062a\u064a\u0639\u0627\u0628\u0647\u0627 \u0641\u064a \u0627\u0644\u0628\u062f\u0627\u064a\u0629 \u0648\u062a\u0643\u0648\u0646 \u0633\u0628\u0628\u064b\u0627 \u0641\u064a \u062f\u0645\u0648\u0639 \u0645\u0639\u0638\u0645 \u0637\u0644\u0627\u0628 \u0627\u0644\u062d\u0633\u0627\u0628 \u0627\u0644\u062a\u0641\u0627\u0636\u0644\u064a \u062d\u0648\u0644 \u0627\u0644\u0639\u0627\u0644\u0645\u060c \u064a\u0645\u0643\u0646 \u062a\u0644\u062e\u064a\u0635\u0647\u0627 \u0645\u0646 \u062e\u0644\u0627\u0644 \u0627\u0644\u0635\u064a\u063a\u0629 \u0627\u0644\u062a\u0627\u0644\u064a\u0629:<\/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>\u062e\u0635\u0627\u0626\u0635 \u0627\u0644\u062d\u062f\u0648\u062f<\/h2>\n<p style=\"text-align: justify;\">\u0623\u0647\u0645\u064a\u0629 \u0627\u0644\u062d\u0635\u0648\u0644 \u0639\u0644\u0649 \u062a\u0639\u0631\u064a\u0641 \u0631\u0633\u0645\u064a \u0644\u0644\u062d\u062f \u0647\u0648 \u0623\u0646\u0647\u060c \u0628\u0646\u0627\u0621\u064b \u0639\u0644\u064a\u0647\u060c \u064a\u0645\u0643\u0646\u0646\u0627 \u0625\u062b\u0628\u0627\u062a \u062e\u0635\u0627\u0626\u0635\u0647\u060c \u0633\u0648\u0627\u0621 \u062a\u0644\u0643 \u0627\u0644\u062a\u064a \u062a\u0628\u062f\u0648 \u0628\u062f\u064a\u0647\u064a\u0629 \u0623\u0648 \u063a\u064a\u0631 \u0630\u0644\u0643.<\/p>\n<p style=\"text-align: justify;\">\u0642\u0628\u0644 \u0627\u0644\u0645\u062a\u0627\u0628\u0639\u0629\u060c \u0639\u0644\u0649 \u0627\u0644\u0631\u063a\u0645 \u0645\u0646 \u0623\u0646\u0647 \u0644\u064a\u0633 \u0636\u0631\u0648\u0631\u064a\u064b\u0627 \u0628\u0634\u0643\u0644 \u0635\u0627\u0631\u0645\u060c \u064a\u064f\u0648\u0635\u0649 \u0628\u0634\u062f\u0629 \u0628\u0645\u0631\u0627\u062c\u0639\u0629 \u0628\u0639\u0636 \u0645\u0641\u0627\u0647\u064a\u0645 <a href=\"http:\/\/toposuranos.com\/material\/es\/category\/matematica\/logica-matematica\/logica-proposicional\/\" rel=\"noopener\" target=\"_blank\"><strong>\u0627\u0644\u0645\u0646\u0637\u0642 \u0627\u0644\u0631\u064a\u0627\u0636\u064a<\/strong><\/a> \u0644\u0641\u0647\u0645 \u0625\u062b\u0628\u0627\u062a\u0627\u062a \u0627\u0644\u062a\u0627\u0644\u064a\u0629 \u0628\u0633\u0647\u0648\u0644\u0629 \u0623\u0643\u0628\u0631.<\/p>\n<p><a name=\"5\"><\/a><\/p>\n<h3>\u0625\u0630\u0627 \u0643\u0627\u0646 \u0627\u0644\u062d\u062f \u0645\u0648\u062c\u0648\u062f\u064b\u0627\u060c \u0641\u0625\u0646\u0647 \u064a\u0643\u0648\u0646 \u0648\u062d\u064a\u062f\u064b\u0627<\/h3>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=FEPfoAfPsFY&amp;t=904s\" rel=\"noopener\" target=\"_blank\"><strong>\u0644\u0625\u062b\u0628\u0627\u062a \u0647\u0630\u0647 \u0627\u0644\u062e\u0627\u0635\u064a\u0629\u060c \u0633\u0646\u0633\u062a\u062e\u062f\u0645 \u062a\u0642\u0646\u064a\u0629 \u0627\u0644\u0628\u0631\u0647\u0627\u0646 \u0628\u0627\u0644\u062a\u0646\u0627\u0642\u0636.<\/strong><\/a> \u0633\u0646\u0628\u062f\u0623 \u0628\u062a\u0639\u0631\u064a\u0641 \u0645\u062c\u0645\u0648\u0639\u0629 \u0627\u0644\u0641\u0631\u0636\u064a\u0627\u062a \u0627\u0644\u062a\u0627\u0644\u064a\u0629:<\/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;\">\u0628\u0646\u0627\u0621\u064b \u0639\u0644\u0649 \u0647\u0630\u0627\u060c \u064a\u0645\u0643\u0646\u0646\u0627 \u0628\u0646\u0627\u0621 \u0627\u0644\u0628\u0631\u0647\u0627\u0646 \u0627\u0644\u0631\u0633\u0645\u064a \u0627\u0644\u062a\u0627\u0644\u064a:<\/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>\u0627\u0641\u062a\u0631\u0627\u0636<\/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>; \u0627\u0641\u062a\u0631\u0627\u0636<\/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>; \u0627\u0641\u062a\u0631\u0627\u0636<\/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>\u0627\u0633\u062a\u062e\u0644\u0627\u0635<\/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>\u0627\u0644\u062a\u0643\u0631\u0627\u0631<\/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>; \u0644\u0623\u0646 <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>; \u0628\u0627\u0633\u062a\u062e\u062f\u0627\u0645(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 + \u0644^\\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) \\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>; \u0645\u0646(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>; \u0646\u0641\u0633 \u0627\u0644\u0625\u062c\u0631\u0627\u0621 \u0643\u0645\u0627 \u0641\u064a (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>-\u0627\u0633\u062a\u0646\u062a\u0627\u062c(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>; \u062a\u0639\u0631\u064a\u0641(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>\u0627\u0644\u0646\u0642\u0637\u0629 \u0627\u0644\u0645\u0631\u0643\u0632\u064a\u0629<\/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>\u062a\u0646\u0627\u0642\u0636<\/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;\">\u0645\u0646 \u0647\u0630\u0627 \u0627\u0644\u0628\u0631\u0647\u0627\u0646\u060c \u0646\u0633\u062a\u0646\u062a\u062c \u0623\u0646\u0647 \u0625\u0630\u0627 \u0643\u0627\u0646 \u0647\u0646\u0627\u0643 \u062d\u062f\u064a\u0646\u060c \u0641\u0647\u0645\u0627 \u0645\u062a\u0633\u0627\u0648\u064a\u0627\u0646\u060c \u0648\u0628\u0627\u0644\u062a\u0627\u0644\u064a \u064a\u0643\u0648\u0646 \u0627\u0644\u062d\u062f \u0648\u062d\u064a\u062f\u064b\u0627.<\/p>\n<p><a name=\"6\"><\/a><\/p>\n<h3>\u062c\u0628\u0631 \u0627\u0644\u062d\u062f\u0648\u062f<\/h3>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=FEPfoAfPsFY&amp;t=2011s\" rel=\"noopener\" target=\"_blank\"><strong>\u0628\u0645\u0627 \u0623\u0646\u0646\u0627 \u0627\u0633\u062a\u0639\u0631\u0636\u0646\u0627 \u0627\u0644\u0623\u0633\u0627\u0633\u064a\u0627\u062a \u062d\u0648\u0644 \u0641\u0643\u0631\u0629 \u0627\u0644\u062d\u062f \u0627\u0644\u0631\u064a\u0627\u0636\u064a.<\/strong><\/a> \u0648\u0644\u0643\u0646 \u0647\u0630\u0627 \u0648\u062d\u062f\u0647 \u0644\u0627 \u064a\u0643\u0641\u064a \u0644\u0625\u062c\u0631\u0627\u0621 \u0627\u0644\u062d\u0633\u0627\u0628\u0627\u062a \u0628\u0627\u0644\u062d\u062f\u0648\u062f\u060c \u0648\u0628\u0627\u0644\u062a\u0623\u0643\u064a\u062f \u0644\u0646 \u064a\u0633\u062a\u062e\u062f\u0645 \u0623\u064a \u0634\u062e\u0635 \u062a\u0639\u0631\u064a\u0641 \u0627\u0644\u062d\u062f \u0644\u0647\u0630\u0627 \u0627\u0644\u063a\u0631\u0636. \u0644\u062d\u0644 \u0647\u0630\u0647 \u0627\u0644\u0645\u0634\u0643\u0644\u0629\u060c \u0633\u0646\u0639\u0645\u0644 \u0627\u0644\u0622\u0646 \u0639\u0644\u0649 \u0627\u0644\u062a\u0642\u0646\u064a\u0627\u062a \u0627\u0644\u062a\u064a \u0633\u062a\u0633\u0627\u0639\u062f\u0646\u0627 \u0639\u0644\u0649 \u0627\u0644\u0628\u062f\u0621 \u0641\u064a \u062d\u0633\u0627\u0628 \u0628\u0639\u0636 \u0627\u0644\u062d\u062f\u0648\u062f.<\/p>\n<p style=\"text-align: justify;\">\u0644\u0646\u0641\u0631\u0636 \u0623\u0646 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x_0, \\alpha, \\beta, L, M \\in \\mathbb{R},<\/span><\/span> \u0648\u0623\u0646 \u062a\u0643\u0648\u0646 f \u0648g \u062f\u0648\u0627\u0644 \u062d\u0642\u064a\u0642\u064a\u0629 \u0628\u062d\u064a\u062b:<\/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;\">\u0625\u0630\u0646\u060c \u062a\u062a\u062d\u0642\u0642 \u0627\u0644\u062e\u0635\u0627\u0626\u0635 \u0627\u0644\u062a\u0627\u0644\u064a\u0629:<\/p>\n<h4>\u062d\u062f \u0645\u062c\u0645\u0648\u0639 \u0623\u0648 \u0641\u0631\u0642 \u062f\u0627\u0644\u062a\u064a\u0646<\/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>\u0625\u062b\u0628\u0627\u062a:<\/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>\u0644\u0646\u0639\u062a\u0628\u0631 \u0645\u062c\u0645\u0648\u0639\u0629 \u0627\u0644\u0641\u0631\u0636\u064a\u0627\u062a<\/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>\u060c \u0648\u0645\u0646 \u0647\u0630\u0647 \u0627\u0644\u0641\u0631\u0636\u064a\u0627\u062a \u064a\u0645\u0643\u0646\u0646\u0627 \u0628\u0646\u0627\u0621 \u0627\u0644\u0627\u0633\u062a\u062f\u0644\u0627\u0644 \u0627\u0644\u062a\u0627\u0644\u064a:<\/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>; \u0627\u0641\u062a\u0631\u0627\u0636<\/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>; \u062a\u0639\u0631\u064a\u0641<\/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>; \u0645\u0646(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>; \u0627\u0641\u062a\u0631\u0627\u0636<\/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>; \u0645\u0634\u0627\u0628\u0647 \u0644\u0640(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>; \u0645\u0646(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>; \u0645\u062a\u0628\u0627\u064a\u0646\u0629 \u0627\u0644\u0645\u062b\u0644\u062b: <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>; \u0645\u0646(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>; \u062a\u0639\u0631\u064a\u0641<\/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>; \u0645\u0646(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>; \u062a\u0639\u0631\u064a\u0641<\/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>; \u0645\u0634\u0627\u0628\u0647\u0629(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>; \u0645\u0646(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>; \u0645\u0646(10,13) <\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>>\u00bb<\/p>\n<h4>\u062d\u062f \u062d\u0627\u0635\u0644 \u0636\u0631\u0628 \u0627\u0644\u062f\u0648\u0627\u0644<\/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>\u0647\u0630\u0627 \u0627\u0644\u0625\u062b\u0628\u0627\u062a \u0623\u0635\u0639\u0628 \u0642\u0644\u064a\u0644\u0627\u064b \u0645\u0646 \u0627\u0644\u0633\u0627\u0628\u0642\u060c<\/strong><\/a> \u0644\u0643\u0646\u0647 \u0644\u064a\u0633 \u0634\u064a\u0626\u0627\u064b \u0644\u0627 \u064a\u0645\u0643\u0646\u0646\u0627 \u062d\u0644\u0647 \u0628\u0628\u0639\u0636 \u0627\u0644\u062d\u064a\u0644 \u0627\u0644\u0628\u0633\u064a\u0637\u0629. \u0628\u0627\u0633\u062a\u062e\u062f\u0627\u0645 \u0646\u0641\u0633 \u0645\u062c\u0645\u0648\u0639\u0629 \u0627\u0644\u0641\u0631\u0636\u064a\u0627\u062a <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathcal{H}<\/span><\/span> \u0645\u0646 \u0627\u0644\u0625\u062b\u0628\u0627\u062a \u0627\u0644\u0633\u0627\u0628\u0642\u060c \u064a\u0645\u0643\u0646\u0646\u0627 \u0628\u0646\u0627\u0621 \u0627\u0644\u0627\u0633\u062a\u062f\u0644\u0627\u0644 \u0627\u0644\u062a\u0627\u0644\u064a:<\/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>; \u062a\u0639\u0631\u064a\u0641<\/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>; \u0627\u0641\u062a\u0631\u0627\u0636<\/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>; \u0628\u0627\u0633\u062a\u062e\u062f\u0627\u0645 (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>; \u062a\u0639\u0631\u064a\u0641<\/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>; \u0627\u0641\u062a\u0631\u0627\u0636<\/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>; \u0628\u0627\u0633\u062a\u062e\u062f\u0627\u0645 (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>; \u0645\u062a\u0628\u0627\u064a\u0646\u0629 \u0627\u0644\u0645\u062b\u0644\u062b + \u062d\u0627\u0644\u0629 \u062e\u0627\u0635\u0629 \u0645\u0646 <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>; \u0645\u0646(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>; \u0645\u062a\u0628\u0627\u064a\u0646\u0629 \u0627\u0644\u0645\u062b\u0644\u062b + \u062d\u0627\u0644\u0629 \u062e\u0627\u0635\u0629 \u0645\u0646 <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>; \u0645\u0646(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>; \u062c\u0645\u0639 \u0635\u0641\u0631<\/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>; \u0627\u0644\u062a\u062d\u0644\u064a\u0644<\/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>; \u0645\u062a\u0628\u0627\u064a\u0646\u0629 \u0627\u0644\u0645\u062b\u0644\u062b(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>; \u0645\u0646(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>; \u0645\u0646(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>; \u0645\u0646(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>; \u0645\u0646(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>; \u0645\u0646(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>\u062d\u062f \u0627\u0644\u062f\u0627\u0644\u0629 \u0627\u0644\u062b\u0627\u0628\u062a\u0629<\/h4>\n<p style=\"text-align: justify; \"><a href=\"https:\/\/www.youtube.com\/watch?v=FEPfoAfPsFY&amp;t=3450s\" rel=\"noopener\" target=\"_blank\"><strong>\u062d\u062f \u0627\u0644\u062f\u0627\u0644\u0629 \u0627\u0644\u062b\u0627\u0628\u062a\u0629<\/strong><\/a> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f(x)=c<\/span><\/span> \u0647\u0648 \u0627\u0644\u062b\u0627\u0628\u062a <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">c<\/span><\/span>. \u0628\u0645\u0639\u0646\u0649<\/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>\u0627\u0644\u0628\u0631\u0647\u0627\u0646<\/strong><\/p>\n<p style=\"text-align: justify; \">\u0627\u0644\u0628\u0631\u0647\u0627\u0646 \u0639\u0644\u0649 \u0647\u0630\u0627 \u0641\u064a \u0627\u0644\u0648\u0627\u0642\u0639 \u0628\u0633\u064a\u0637\u060c \u0644\u0623\u0646\u0647 \u0641\u064a \u0627\u0644\u062d\u0642\u064a\u0642\u0629 \u064a\u062a\u0639\u0644\u0642 \u0628\u062a\u0639\u0631\u064a\u0641 \u0628\u062f\u064a\u0647\u064a. \u0645\u0646 \u0627\u0644\u0645\u0639\u0631\u0648\u0641 \u0623\u0646:<\/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; \">\u0644\u0643\u0646 \u064a\u062d\u062f\u062b \u0623\u0646 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0=|c-c|\\lt \\epsilon<\/span><\/span> \u0647\u0648 \u062a\u0639\u0631\u064a\u0641 \u0628\u062f\u064a\u0647\u064a \u0644\u0623\u064a \u0642\u064a\u0645\u0629 \u0645\u0648\u062c\u0628\u0629 \u0644\u0640 \u00ab\u0625\u0628\u0633\u064a\u0644\u0648\u0646\u00bb\u060c \u0648\u0628\u0627\u0644\u062a\u0627\u0644\u064a \u062a\u0643\u0648\u0646 \u0627\u0644\u0646\u062a\u064a\u062c\u0629 \u0623\u064a\u0636\u064b\u0627 \u0628\u062f\u064a\u0647\u064a\u0629\u060c \u0648\u0628\u0646\u0627\u0621\u064b \u0639\u0644\u0649 \u0630\u0644\u0643\u060c \u0627\u0644\u062a\u0639\u0628\u064a\u0631 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to x_0}c = c <\/span><\/span> \u0647\u0648 \u0623\u064a\u0636\u064b\u0627 \u0628\u062f\u064a\u0647\u064a.<\/p>\n<h4>\u062d\u062f \u0646\u0627\u062a\u062c \u0642\u0633\u0645\u0629 \u0627\u0644\u062f\u0648\u0627\u0644<\/h4>\n<p style=\"text-align: justify; \"><a href=\"https:\/\/www.youtube.com\/watch?v=FEPfoAfPsFY&amp;t=3563s\" rel=\"noopener\" target=\"_blank\"><strong>\u0627\u0644\u0622\u0646 \u0646\u062d\u0646 \u0641\u064a \u0648\u0636\u0639 \u064a\u0645\u0643\u0646\u0646\u0627 \u0645\u0646 \u0625\u062b\u0628\u0627\u062a \u0642\u0627\u0639\u062f\u0629 \u062d\u062f \u0646\u0627\u062a\u062c \u0642\u0633\u0645\u0629 \u062f\u0627\u0644\u062a\u064a\u0646.<\/strong><\/a> \u0647\u0630\u0647 \u0627\u0644\u0642\u0627\u0639\u062f\u0629 \u0647\u064a<\/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; \">\u062d\u064a\u062b\u060c \u0645\u062b\u0644\u0645\u0627 \u0647\u0648 \u0627\u0644\u062d\u0627\u0644 \u0645\u0639 \u0627\u0644\u062e\u0635\u0627\u0626\u0635 \u0627\u0644\u0633\u0627\u0628\u0642\u0629\u060c \u0646\u0641\u062a\u0631\u0636 \u0623\u0646 \u0645\u062c\u0645\u0648\u0639\u0629 \u0627\u0644\u0641\u0631\u0636\u064a\u0627\u062a<\/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>\u0627\u0644\u0628\u0631\u0647\u0627\u0646<\/strong><\/p>\n<p style=\"text-align: justify; \">\u0644\u062d\u0633\u0646 \u0627\u0644\u062d\u0638\u060c \u0644\u0646 \u0646\u062d\u062a\u0627\u062c \u0625\u0644\u0649 \u0625\u062c\u0631\u0627\u0621 \u0627\u0644\u0645\u0632\u064a\u062f \u0645\u0646 \u0627\u0644\u0628\u0631\u0627\u0647\u064a\u0646 \u0645\u062b\u0644 \u0627\u0644\u062a\u064a \u0642\u0645\u0646\u0627 \u0628\u0647\u0627 \u0633\u0627\u0628\u0642\u064b\u0627\u060c \u0644\u0623\u0646\u0646\u0627 \u064a\u0645\u0643\u0646\u0646\u0627 \u0627\u0644\u0622\u0646 \u0627\u0633\u062a\u062e\u062f\u0627\u0645 \u0647\u0630\u0647 \u0627\u0644\u0646\u062a\u0627\u0626\u062c \u0645\u0628\u0627\u0634\u0631\u0629 \u0644\u062a\u062d\u0642\u064a\u0642 \u0623\u0647\u062f\u0627\u0641\u0646\u0627. \u0648\u0644\u0643\u0646 \u0642\u0628\u0644 \u0630\u0644\u0643\u060c \u062f\u0639\u0648\u0646\u0627 \u0646\u062b\u0628\u062a \u0623\u0648\u0644\u0627\u064b \u0623\u0646<\/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; \">\u0644\u0625\u062b\u0628\u0627\u062a \u0630\u0644\u0643\u060c \u064a\u0643\u0641\u064a \u0627\u0633\u062a\u062e\u062f\u0627\u0645 \u0642\u0627\u0639\u062f\u0629 \u062d\u062f \u0646\u0627\u062a\u062c \u062d\u0627\u0635\u0644 \u0636\u0631\u0628 \u0648\u062d\u062f \u0627\u0644\u062f\u0627\u0644\u0629 \u0627\u0644\u062b\u0627\u0628\u062a\u0629 \u0645\u062c\u062a\u0645\u0639\u0629\u060c \u0641\u0642\u0637 \u0639\u0644\u064a\u0646\u0627 \u0623\u0646 \u0646\u0643\u0648\u0646 \u062d\u0630\u0631\u064a\u0646 \u0641\u064a \u0623\u0646 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">g(x)<\/span><\/span> \u0644\u0627 \u062a\u0643\u0648\u0646 \u0635\u0641\u0631\u064b\u0627:<\/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; \">\u0648\u0628\u0627\u0644\u062a\u0627\u0644\u064a: <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; \">\u0648\u0623\u062e\u064a\u0631\u0627\u064b\u060c \u0645\u0646 \u062e\u0644\u0627\u0644 \u0642\u0627\u0639\u062f\u0629 \u062d\u062f \u062d\u0627\u0635\u0644 \u0636\u0631\u0628 \u0627\u0644\u062f\u0648\u0627\u0644:<\/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; \">\u0633\u064a\u062a\u062d\u0642\u0642 \u0647\u0630\u0627 \u062f\u0627\u0626\u0645\u064b\u0627 \u0637\u0627\u0644\u0645\u0627 \u0623\u0646 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">M<\/span><\/span> \u0644\u064a\u0633 \u0635\u0641\u0631\u064b\u0627.<\/p>\n<h4>\u062d\u062f \u0627\u0644\u0642\u0648\u0629 \u0627\u0644\u0637\u0628\u064a\u0639\u064a\u0629<\/h4>\n<p style=\"text-align: justify; \"><a href=\"https:\/\/www.youtube.com\/watch?v=FEPfoAfPsFY&amp;t=3725s\" rel=\"noopener\" target=\"_blank\"><strong>\u0647\u0630\u0647 \u0627\u0644\u062e\u0627\u0635\u064a\u0629 \u062a\u0642\u0648\u0644 \u0644\u0646\u0627 \u0623\u0646\u0647\u060c<\/strong><\/a> \u0625\u0630\u0627 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x_0 \\to x_0}f(x) = L<\/span><\/span>\u060c \u0641\u0625\u0646 \u0630\u0644\u0643 \u064a\u0639\u0646\u064a <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>. \u064a\u0645\u0643\u0646\u0646\u0627 \u0625\u062b\u0628\u0627\u062a \u0630\u0644\u0643 \u0628\u0627\u0633\u062a\u062e\u062f\u0627\u0645 \u0627\u0644\u0627\u0633\u062a\u0642\u0631\u0627\u0621 \u0627\u0644\u0631\u064a\u0627\u0636\u064a.<\/p>\n<p style=\"text-align: justify; color: #000080;\"><strong>\u0627\u0644\u0628\u0631\u0647\u0627\u0646:<\/strong><\/p>\n<ul style=\"text-align: justify; \">\n<li><strong>\u0627\u0644\u062d\u0627\u0644\u0629 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">n=1<\/span><\/span>:<\/strong> (\u0627\u0644\u062e\u0637\u0648\u0629 \u0627\u0644\u0623\u0648\u0644\u0649)\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> \u0648\u0647\u0630\u0627 \u064a\u0646\u0647\u064a \u0627\u0644\u062e\u0637\u0648\u0629 \u0627\u0644\u0623\u0648\u0644\u0649 \u2705<\/p>\n<\/li>\n<li><strong>\u0627\u0644\u062d\u0627\u0644\u0629 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">n=k<\/span><\/span>:<\/strong> (\u062e\u0637\u0648\u0629 \u0627\u0644\u0627\u0633\u062a\u0642\u0631\u0627\u0621)\n<p style=\"text-align: justify;\">\u0628\u0627\u0641\u062a\u0631\u0627\u0636 \u0623\u0646 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to x_0} [f(x)]^k = L^k <\/span><\/span> (\u0641\u0631\u0636\u064a\u0629 \u0627\u0644\u0627\u0633\u062a\u0642\u0631\u0627\u0621)\u060c \u0646\u062a\u062d\u0642\u0642 \u0645\u0646 \u0623\u0646 <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> \u0635\u062d\u064a\u062d\u0629.<\/p>\n<p style=\"text-align: justify;\">\u0646\u062c\u062f \u0623\u0646: <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>. \u0647\u0630\u0627 \u0628\u0627\u0633\u062a\u062e\u062f\u0627\u0645 \u0642\u0627\u0639\u062f\u0629 \u062d\u062f \u0646\u0627\u062a\u062c \u062d\u0627\u0635\u0644 \u0636\u0631\u0628 \u0627\u0644\u062f\u0648\u0627\u0644 \u0627\u0644\u0645\u062b\u0628\u062a\u0629 \u0633\u0627\u0628\u0642\u064b\u0627.<\/p>\n<p style=\"text-align: justify;\">\u062b\u0645\u060c \u0628\u0646\u0627\u0621\u064b \u0639\u0644\u0649 \u0641\u0631\u0636\u064a\u0629 \u0627\u0644\u0627\u0633\u062a\u0642\u0631\u0627\u0621\u060c \u0646\u062c\u062f \u0623\u0646 <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> \u0647\u0630\u0627 \u064a\u0646\u0647\u064a \u062e\u0637\u0648\u0629 \u0627\u0644\u0627\u0633\u062a\u0642\u0631\u0627\u0621 \u2705<\/p>\n<\/li>\n<li>\u0648\u0628\u0627\u0644\u062a\u0627\u0644\u064a: <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>\u062d\u062f \u0627\u0644\u062c\u0630\u0631 \u0627\u0644\u0646\u0648\u0646\u064a<\/h4>\n<p style=\"text-align: justify; \"><a href=\"https:\/\/www.youtube.com\/watch?v=FEPfoAfPsFY&amp;t=3912s\" rel=\"noopener\" target=\"_blank\"><strong>\u0628\u0634\u0643\u0644 \u0645\u0634\u0627\u0628\u0647 \u0644\u0644\u0642\u0648\u0629\u060c \u064a\u062a\u062d\u0642\u0642 \u0623\u0646<\/strong><\/a> <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} \\sqrt[n]{f(x)} = \\sqrt[n]{L} \\right) <\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000080;\"><strong>\u0627\u0644\u0628\u0631\u0647\u0627\u0646:<\/strong><\/p>\n<p style=\"text-align: justify; \">\u0628\u0627\u0633\u062a\u062e\u062f\u0627\u0645 \u0642\u0627\u0639\u062f\u0629 \u0627\u0644\u0642\u0648\u0629 \u0627\u0644\u062a\u064a \u0642\u0645\u0646\u0627 \u0628\u0625\u062b\u0628\u0627\u062a\u0647\u0627 \u0645\u0633\u0628\u0642\u064b\u0627 \u0646\u062c\u062f \u0623\u0646<\/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; \">\u0648\u0628\u0627\u0644\u062a\u0627\u0644\u064a: <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>\u062d\u062f \u0627\u0644\u0642\u0648\u0649 \u0627\u0644\u0643\u0633\u0631\u064a\u0629<\/h4>\n<p style=\"text-align: justify; \"><a href=\"https:\/\/www.youtube.com\/watch?v=FEPfoAfPsFY&amp;t=4007s\" rel=\"noopener\" target=\"_blank\"><strong>\u0628\u0627\u0633\u062a\u062e\u062f\u0627\u0645 \u0627\u0644\u0642\u0648\u0627\u0639\u062f \u0627\u0644\u0645\u0633\u062a\u062e\u0644\u0635\u0629 \u0645\u0646 \u0627\u0644\u0628\u0631\u0627\u0647\u064a\u0646 \u0627\u0644\u0633\u0627\u0628\u0642\u0629<\/strong><\/a> \u064a\u0645\u0643\u0646\u0646\u0627 \u0627\u0644\u0648\u0635\u0648\u0644 \u0625\u0644\u0649 \u0627\u0644\u0628\u0631\u0647\u0627\u0646 \u0627\u0644\u0623\u062e\u064a\u0631\u060c \u0648\u0647\u0648: <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>. \u064a\u062a\u0645 \u0627\u0644\u062d\u0635\u0648\u0644 \u0639\u0644\u064a\u0647 \u0628\u0641\u0636\u0644 \u0642\u0627\u0639\u062f\u0629 \u0646\u0627\u062a\u062c \u062d\u0627\u0635\u0644 \u0636\u0631\u0628 \u0627\u0644\u062f\u0648\u0627\u0644 \u0644\u0623\u0646 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle [f(x)]^{\\frac{p}{q}} =[\\sqrt[q]{f(x)}]^p <\/span><\/span> \u0648 <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>\u062d\u062f <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>\u0645\u0639 \u0647\u0630\u0627 \u0627\u0644\u0628\u0631\u0647\u0627\u0646 \u0646\u062e\u062a\u0645 \u0633\u0644\u0633\u0644\u0629 \u0647\u0630\u0647 \u0627\u0644\u0628\u0631\u0627\u0647\u064a\u0646\u060c<\/strong><\/a> \u0648\u0645\u0639 \u0627\u0644\u0628\u0631\u0627\u0647\u064a\u0646 \u0627\u0644\u0633\u0627\u0628\u0642\u0629 \u0633\u0646\u0643\u0648\u0646 \u0642\u0627\u062f\u0631\u064a\u0646 \u0645\u0646 \u0627\u0644\u0622\u0646 \u0641\u0635\u0627\u0639\u062f\u064b\u0627 \u0639\u0644\u0649 \u062d\u0633\u0627\u0628 \u0627\u0644\u0639\u062f\u064a\u062f \u0645\u0646 \u0627\u0644\u062d\u062f\u0648\u062f \u0628\u0634\u0643\u0644 \u0634\u0628\u0647 \u0628\u062f\u064a\u0647\u064a.<\/p>\n<p style=\"text-align: justify; \">\u0645\u0646 \u0627\u0644\u0633\u0647\u0644 \u0625\u062b\u0628\u0627\u062a \u0623\u0646 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to x_0}x = x_0<\/span><\/span>\u060c \u0644\u0623\u0646\u0647 \u0644\u0643\u064a \u064a\u062a\u062d\u0642\u0642 \u0647\u0630\u0627\u060c \u064a\u062c\u0628 \u0623\u0646:<\/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; \">\u0648\u0641\u0642\u064b\u0627 \u0644\u062a\u0639\u0631\u064a\u0641 \u0627\u0644\u062d\u062f\u060c \u064a\u062c\u0628 \u0623\u0646 \u064a\u0643\u0648\u0646 \u0647\u0646\u0627\u0643 \u062f\u0627\u0626\u0645\u064b\u0627 \u0639\u0644\u0649 \u0627\u0644\u0623\u0642\u0644 \u062f\u0644\u062a\u0627 \u0648\u0627\u062d\u062f\u0629 \u0644\u0623\u064a \u0642\u064a\u0645\u0629 \u0625\u0628\u0633\u064a\u0644\u0648\u0646 \u0628\u062d\u064a\u062b \u062a\u062d\u0642\u0642 \u062c\u0645\u064a\u0639 \u0627\u0644\u0634\u0631\u0648\u0637 \u0627\u0644\u0623\u062e\u0631\u0649\u061b \u0648\u0628\u0630\u0644\u0643 \u064a\u0643\u0641\u064a \u0625\u064a\u062c\u0627\u062f \u0648\u0627\u062d\u062f\u0629 \u0644\u0644\u062a\u062d\u0642\u0642 \u0645\u0646 \u0623\u0646 \u0627\u0644\u062d\u062f \u0647\u0648 \u0643\u0645\u0627 \u064a\u064f\u0641\u062a\u0631\u0636 \u0623\u0646 \u064a\u0643\u0648\u0646. \u0644\u0643\u0646 \u0647\u0630\u0627 \u0641\u064a \u0627\u0644\u0648\u0627\u0642\u0639 \u0623\u0645\u0631 \u0648\u0627\u0636\u062d\u060c \u0644\u0623\u0646\u0647 \u064a\u0643\u0641\u064a \u0645\u0644\u0627\u062d\u0638\u0629 \u0623\u0646 \u0623\u064a <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\delta\\leq\\epsilon<\/span><\/span> \u064a\u062d\u0642\u0642 \u0647\u0630\u0627 \u0627\u0644\u0634\u0631\u0637. \u0648\u0628\u0627\u0644\u062a\u0627\u0644\u064a: <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>\u062d\u0633\u0627\u0628 \u0627\u0644\u062d\u062f\u0648\u062f \u0627\u0644\u0628\u0633\u064a\u0637\u0629<\/h2>\n<p style=\"text-align: justify; \"><a href=\"https:\/\/www.youtube.com\/watch?v=FEPfoAfPsFY&amp;t=4155s\" rel=\"noopener\" target=\"_blank\"><strong>\u0628\u0641\u0636\u0644 \u062c\u0645\u064a\u0639 \u0647\u0630\u0647 \u0627\u0644\u0646\u0638\u0631\u064a\u0627\u062a \u0627\u0644\u062a\u064a \u0642\u0645\u0646\u0627 \u0628\u0645\u0631\u0627\u062c\u0639\u062a\u0647\u0627 \u0644\u0644\u062a\u0648<\/strong><\/a> \u064a\u0645\u0643\u0646 \u062d\u0633\u0627\u0628 \u0645\u062c\u0645\u0648\u0639\u0629 \u0643\u0628\u064a\u0631\u0629 \u0645\u0646 \u0627\u0644\u062d\u062f\u0648\u062f \u0628\u0637\u0631\u064a\u0642\u0629 \u0628\u062f\u064a\u0647\u064a\u0629\u060c \u0643\u0645\u0627 \u0644\u0648 \u0623\u0646\u0646\u0627 \u0646\u0642\u0648\u0645 \u0641\u0642\u0637 \u0628\u062a\u0642\u064a\u064a\u0645 \u0627\u0644\u062f\u0627\u0644\u0629. \u0647\u0646\u0627 \u064a\u0645\u0643\u0646\u0643 \u0645\u0634\u0627\u0647\u062f\u0629 \u0628\u0639\u0636 \u0627\u0644\u0623\u0645\u062b\u0644\u0629:<\/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>\u062d\u062f\u0648\u062f \u0627\u0644\u062f\u0648\u0627\u0644 \u0630\u0627\u062a \u0627\u0644\u0645\u062a\u063a\u064a\u0631 \u0627\u0644\u062d\u0642\u064a\u0642\u064a \u0645\u0644\u062e\u0635: \u0641\u064a \u0647\u0630\u0647 \u0627\u0644\u062d\u0635\u0629\u060c \u0633\u0646\u0642\u0648\u0645 \u0628\u0645\u0631\u0627\u062c\u0639\u0629 \u0634\u0627\u0645\u0644\u0629 \u0644\u0644\u062a\u0639\u0631\u064a\u0641 \u0627\u0644\u0631\u0633\u0645\u064a \u0644\u062d\u062f\u0648\u062f \u0627\u0644\u062f\u0648\u0627\u0644 \u0630\u0627\u062a \u0627\u0644\u0645\u062a\u063a\u064a\u0631 \u0627\u0644\u062d\u0642\u064a\u0642\u064a\u060c \u0648\u0645\u0646 \u062e\u0644\u0627\u0644 \u0630\u0644\u0643 \u0633\u0646\u062b\u0628\u062a \u0627\u0644\u062e\u0635\u0627\u0626\u0635 \u0627\u0644\u0631\u0626\u064a\u0633\u064a\u0629 \u0627\u0644\u062a\u064a \u062a\u0642\u0648\u062f \u0625\u0644\u0649 \u062c\u0628\u0631 \u0627\u0644\u062d\u062f\u0648\u062f. \u0623\u0647\u062f\u0627\u0641 \u0627\u0644\u062a\u0639\u0644\u0645: \u0641\u064a \u0646\u0647\u0627\u064a\u0629 \u0647\u0630\u0647 \u0627\u0644\u062d\u0635\u0629\u060c \u0633\u064a\u0643\u0648\u0646 \u0627\u0644\u0637\u0627\u0644\u0628 \u0642\u0627\u062f\u0631\u064b\u0627 \u0639\u0644\u0649: \u062a\u0630\u0643\u0631 \u062a\u0639\u0631\u064a\u0641 \u062d\u062f \u0627\u0644\u062f\u0648\u0627\u0644 \u0630\u0627\u062a \u0627\u0644\u0645\u062a\u063a\u064a\u0631 \u0627\u0644\u062d\u0642\u064a\u0642\u064a. \u0625\u062b\u0628\u0627\u062a \u0627\u0644\u062e\u0635\u0627\u0626\u0635 \u0627\u0644\u062a\u064a \u062a\u0642\u0648\u062f \u0625\u0644\u0649 \u062c\u0628\u0631 \u0627\u0644\u062d\u062f\u0648\u062f \u0645\u0646 [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":27786,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"iawp_total_views":1,"footnotes":""},"categories":[860,565],"tags":[],"class_list":["post-27827","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-860","category-565"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v26.7 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>\u062d\u062f \u0627\u0644\u062f\u0648\u0627\u0644 \u0644\u0645\u062a\u063a\u064a\u0631 \u062d\u0642\u064a\u0642\u064a \u0648\u0627\u062d\u062f - toposuranos.com\/material<\/title>\n<meta name=\"description\" content=\"\u0641\u064a \u062d\u0633\u0627\u0628 \u0627\u0644\u062a\u0641\u0627\u0636\u0644 \u0648\u0627\u0644\u062a\u0643\u0627\u0645\u0644\u060c \u064a\u0635\u0641 \u062d\u062f \u0627\u0644\u062f\u0648\u0627\u0644 \u0633\u0644\u0648\u0643\u0647\u0627 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