{"id":27837,"date":"2024-08-11T13:00:13","date_gmt":"2024-08-11T13:00:13","guid":{"rendered":"http:\/\/toposuranos.com\/material\/?p=27837"},"modified":"2024-09-12T20:34:32","modified_gmt":"2024-09-12T20:34:32","slug":"%e0%a4%b5%e0%a4%be%e0%a4%b8%e0%a5%8d%e0%a4%a4%e0%a4%b5%e0%a4%bf%e0%a4%95-%e0%a4%9a%e0%a4%b0-%e0%a4%95%e0%a5%87-%e0%a4%8f%e0%a4%95-%e0%a4%ab%e0%a4%bc%e0%a4%82%e0%a4%95%e0%a5%8d%e0%a4%b6%e0%a4%a8","status":"publish","type":"post","link":"http:\/\/toposuranos.com\/material\/hi\/%e0%a4%b5%e0%a4%be%e0%a4%b8%e0%a5%8d%e0%a4%a4%e0%a4%b5%e0%a4%bf%e0%a4%95-%e0%a4%9a%e0%a4%b0-%e0%a4%95%e0%a5%87-%e0%a4%8f%e0%a4%95-%e0%a4%ab%e0%a4%bc%e0%a4%82%e0%a4%95%e0%a5%8d%e0%a4%b6%e0%a4%a8\/","title":{"rendered":"\u0935\u093e\u0938\u094d\u0924\u0935\u093f\u0915 \u091a\u0930 \u0915\u0947 \u090f\u0915 \u092b\u093c\u0902\u0915\u094d\u0936\u0928 \u0915\u0940 \u0938\u0940\u092e\u093e"},"content":{"rendered":"<p><center><\/p>\n<h1>\u0935\u093e\u0938\u094d\u0924\u0935\u093f\u0915 \u091a\u0930 \u0935\u093e\u0932\u0947 \u0915\u093e\u0930\u094d\u092f\u094b\u0902 \u0915\u0940 \u0938\u0940\u092e\u093e<\/h1>\n<p><\/center><\/p>\n<p style=\"text-align:center\"><em><strong>\u0938\u093e\u0930\u093e\u0902\u0936:<\/strong><br \/>\n\u0907\u0938 \u0915\u0915\u094d\u0937\u093e \u092e\u0947\u0902 \u090f\u0915 \u0935\u093e\u0938\u094d\u0924\u0935\u093f\u0915 \u091a\u0930 \u0935\u093e\u0932\u0947 \u0915\u093e\u0930\u094d\u092f\u094b\u0902 \u0915\u0940 \u0938\u0940\u092e\u093e \u0915\u0940 \u0914\u092a\u091a\u093e\u0930\u093f\u0915 \u092a\u0930\u093f\u092d\u093e\u0937\u093e \u0915\u0940 \u0917\u0939\u0930\u093e\u0908 \u0938\u0947 \u0938\u092e\u0940\u0915\u094d\u0937\u093e \u0915\u0940 \u091c\u093e\u0924\u0940 \u0939\u0948, \u0914\u0930 \u0907\u0938\u0915\u0947 \u0906\u0927\u093e\u0930 \u092a\u0930 \u0938\u0940\u092e\u093e\u0913\u0902 \u0915\u0947 \u092c\u0940\u091c\u0917\u0923\u093f\u0924 \u0915\u0940 \u092a\u094d\u0930\u092e\u0941\u0916 \u0917\u0941\u0923\u094b\u0902 \u0915\u094b \u092a\u094d\u0930\u0926\u0930\u094d\u0936\u093f\u0924 \u0915\u093f\u092f\u093e \u091c\u093e\u0924\u093e \u0939\u0948\u0964<\/br><\/em><\/p>\n<p style=\"text-align:center\"><em><strong>\u0936\u093f\u0915\u094d\u0937\u0923 \u0909\u0926\u094d\u0926\u0947\u0936\u094d\u092f:<\/strong><br \/>\n\u0907\u0938 \u0915\u0915\u094d\u0937\u093e \u0915\u0947 \u0905\u0902\u0924 \u092e\u0947\u0902 \u091b\u093e\u0924\u094d\u0930 \u0938\u0915\u094d\u0937\u092e \u0939\u094b\u0917\u093e:<\/p>\n<ul>\n<li>\u090f\u0915 \u0935\u093e\u0938\u094d\u0924\u0935\u093f\u0915 \u091a\u0930 \u0935\u093e\u0932\u0947 \u0915\u093e\u0930\u094d\u092f\u094b\u0902 \u0915\u0940 \u0938\u0940\u092e\u093e \u0915\u0940 <strong>\u092f\u093e\u0926<\/strong> \u0926\u093f\u0932\u093e\u0928\u093e\u0964<\/li>\n<li><strong>\u0938\u093f\u0926\u094d\u0927<\/strong> \u0915\u0930\u0928\u093e \u0915\u093f <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\epsilon-\\delta<\/span><\/span> \u0915\u0940 \u0938\u0939\u093e\u092f\u0924\u093e \u0938\u0947 \u0938\u0940\u092e\u093e\u0913\u0902 \u0915\u0947 \u092c\u0940\u091c\u0917\u0923\u093f\u0924 \u0915\u094b \u0915\u0948\u0938\u0947 \u0938\u093f\u0926\u094d\u0927 \u0915\u093f\u092f\u093e \u091c\u093e \u0938\u0915\u0924\u093e \u0939\u0948\u0964<\/li>\n<li>\u0938\u0940\u092e\u093e\u0913\u0902 \u0915\u0947 \u092c\u0940\u091c\u0917\u0923\u093f\u0924 \u0914\u0930 \u0907\u0938\u0915\u0940 \u0917\u0941\u0923\u094b\u0902 \u0915\u093e \u0909\u092a\u092f\u094b\u0917 \u0915\u0930\u0915\u0947 \u0935\u093e\u0938\u094d\u0924\u0935\u093f\u0915 \u091a\u0930 \u0935\u093e\u0932\u0947 \u0915\u093e\u0930\u094d\u092f\u094b\u0902 \u0915\u0940 \u0938\u0940\u092e\u093e\u0913\u0902 \u0915\u0940 <strong>\u0917\u0923\u0928\u093e<\/strong> \u0915\u0930\u0928\u093e\u0964<\/li>\n<\/ul>\n<p><\/em><\/p>\n<p><center><br \/>\n<strong>\u0935\u093f\u0937\u092f-\u0938\u0942\u091a\u0940<\/strong><br \/>\n<a href=\"#1\"><strong>\u092a\u0930\u093f\u091a\u092f<\/strong><\/a><br \/>\n<a href=\"#2\"><strong>\u090f\u0915 \u0917\u094d\u0930\u093e\u092b\u093f\u0915 \u0926\u0943\u0937\u094d\u091f\u093f\u0915\u094b\u0923 \u0938\u0947 \u090f\u0915 \u0915\u093e\u0930\u094d\u092f \u0915\u0940 \u0938\u0940\u092e\u093e \u0915\u0940 \u0938\u0939\u091c \u0927\u093e\u0930\u0923<\/strong><\/a><br \/>\n<a href=\"#3\"><strong>\u0938\u0940\u092e\u093e \u0915\u0940 \u0914\u092a\u091a\u093e\u0930\u093f\u0915 \u092a\u0930\u093f\u092d\u093e\u0937\u093e<\/strong><\/a><br \/>\n<a href=\"#4\"><strong>\u0938\u0940\u092e\u093e\u0913\u0902 \u0915\u0947 \u0917\u0941\u0923<\/strong><\/a><br \/>\n<a href=\"#5\">\u092f\u0926\u093f \u0938\u0940\u092e\u093e \u092e\u094c\u091c\u0942\u0926 \u0939\u0948, \u0924\u094b \u092f\u0939 \u0905\u0926\u094d\u0935\u093f\u0924\u0940\u092f \u0939\u0948<\/a><br \/>\n<a href=\"#6\">\u0938\u0940\u092e\u093e\u0913\u0902 \u0915\u093e \u092c\u0940\u091c\u0917\u0923\u093f\u0924<\/a><br \/>\n<a href=\"#7\">\u0938\u0930\u0932 \u0938\u0940\u092e\u093e\u0913\u0902 \u0915\u0940 \u0917\u0923\u0928\u093e<\/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>\u092a\u0930\u093f\u091a\u092f<\/h2>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=FEPfoAfPsFY&amp;t=240s\" rel=\"noopener\" target=\"_blank\"><strong><\u0915\u094d\u092f\u093e \u092c\u0940\u091c\u0917\u0923\u093f\u0924 \u0914\u0930 \u091c\u094d\u092f\u093e\u092e\u093f\u0924\u093f \u0915\u093e \u0905\u0927\u094d\u092f\u092f\u0928 \u0917\u0923\u093f\u0924 \u0938\u0947 \u0915\u0948\u0938\u0947 \u092d\u093f\u0928\u094d\u0928 \u0939\u0948?<\/strong><\/a> \u0907\u0938 \u092a\u094d\u0930\u0936\u094d\u0928 \u0915\u093e \u0909\u0924\u094d\u0924\u0930 \u0939\u092e\u0947\u0902 \u0938\u0940\u092e\u093e \u0915\u0940 \u0905\u0935\u0927\u093e\u0930\u0923\u093e \u0938\u0947 \u092e\u093f\u0932\u0924\u093e \u0939\u0948\u0964 \u0907\u0938 \u0932\u0947\u0916 \u092e\u0947\u0902 \u0938\u0940\u092e\u093e \u0914\u0930 \u0907\u0938\u0915\u0940 \u092a\u0930\u093f\u092d\u093e\u0937\u093e \u0915\u093e \u0905\u0927\u094d\u092f\u092f\u0928 \u0915\u093f\u092f\u093e \u0917\u092f\u093e \u0939\u0948\u0964<\/p>\n<p style=\"text-align: justify;\">\u0938\u0940\u092e\u093e \u0936\u092c\u094d\u0926 \u0915\u094b \u0906\u092e\u0924\u094c\u0930 \u092a\u0930 \u0939\u092e \u0915\u093f\u0938\u0940 \u092a\u094d\u0930\u0915\u093e\u0930 \u0915\u0940 \u0938\u0940\u092e\u093e \u0938\u0947 \u091c\u094b\u0921\u093c\u0924\u0947 \u0939\u0948\u0902, \u091c\u0948\u0938\u0947 \u0915\u093f \u0915\u093f\u0938\u0940 \u0905\u0902\u0924\u0930\u093e\u0932 \u0915\u0947 \u0938\u093f\u0930\u094b\u0902 a, b \u0915\u0940 \u0938\u0940\u092e\u093e (\u0907\u0928\u0915\u0940 \u092a\u094d\u0930\u0915\u0943\u0924\u093f \u0915\u0940 \u092a\u0930\u0935\u093e\u0939 \u0915\u093f\u090f \u092c\u093f\u0928\u093e)<\/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;\">\u092f\u093e \u0935\u0930\u094d\u0924\u092e\u093e\u0928 \u0915\u0940 \u0924\u0930\u0939, \u091c\u093f\u0938\u0947 \u0939\u092e \u0915\u0939 \u0938\u0915\u0924\u0947 \u0939\u0948\u0902 \u0915\u093f \u092f\u0939 \u0905\u0924\u0940\u0924 \u0914\u0930 \u092d\u0935\u093f\u0937\u094d\u092f \u0915\u0947 \u092c\u0940\u091a \u0915\u0940 \u0938\u0940\u092e\u093e \u0939\u0948\u0964 \u0907\u0938\u0940 \u0924\u0930\u0939, \u0938\u0940\u092e\u093e \u0915\u0940 \u0905\u0935\u0927\u093e\u0930\u0923\u093e \u0939\u092e\u0947\u0902 \u0907\u0938 \u0938\u0939\u091c \u0927\u093e\u0930\u0923\u093e \u0915\u094b \u0917\u0923\u093f\u0924\u0940\u092f \u0930\u0942\u092a \u092e\u0947\u0902 \u0938\u092e\u091d\u0928\u0947 \u0915\u0940 \u0905\u0928\u0941\u092e\u0924\u093f \u0926\u0947\u0924\u0940 \u0939\u0948 \u0915\u093f \u0939\u092e \u0915\u093f\u0938\u0940 \u0935\u093f\u0936\u0947\u0937 \u092c\u093f\u0902\u0926\u0941 \u0915\u0947 \u091c\u093f\u0924\u0928\u093e \u091a\u093e\u0939\u0947\u0902 \u0909\u0924\u0928\u093e \u0915\u0930\u0940\u092c \u092a\u0939\u0941\u0901\u091a \u0938\u0915\u0924\u0947 \u0939\u0948\u0902\u0964<\/p>\n<p><a name=\"2\"><\/a><\/p>\n<h2>\u090f\u0915 \u0917\u094d\u0930\u093e\u092b\u093f\u0915 \u0926\u0943\u0937\u094d\u091f\u093f\u0915\u094b\u0923 \u0938\u0947 \u090f\u0915 \u0915\u093e\u0930\u094d\u092f \u0915\u0940 \u0938\u0940\u092e\u093e \u0915\u0940 \u0938\u0939\u091c \u0927\u093e\u0930\u0923<\/h2>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=FEPfoAfPsFY&amp;t=314s\" rel=\"noopener\" target=\"_blank\"><strong>\u0938\u0940\u092e\u093e \u0915\u0940 \u0927\u093e\u0930\u0923\u093e \u0915\u094b \u0938\u092e\u091d\u0928\u0947 \u0915\u0947 \u0932\u093f\u090f, \u092f\u0939 \u090f\u0915 \u0917\u094d\u0930\u093e\u092b\u093f\u0915 \u092a\u094d\u0930\u0924\u093f\u0928\u093f\u0927\u093f\u0924\u094d\u0935 \u0938\u0947 \u0936\u0941\u0930\u0942 \u0915\u0930\u0928\u093e \u0909\u091a\u093f\u0924 \u0939\u0948<\/strong><\/a> \u0914\u0930 \u092a\u0942\u091b\u0928\u093e \u0915\u093f \u091c\u092c <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f(x)<\/span><\/span> \u0915\u0947 \u0938\u093e\u0925 \u0915\u094d\u092f\u093e \u0939\u094b\u0924\u093e \u0939\u0948 \u091c\u092c <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x<\/span><\/span> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x_0<\/span><\/span> \u0915\u0947 \u091c\u093f\u0924\u0928\u093e \u0915\u0930\u0940\u092c \u0939\u094b \u0938\u0915\u0947 \u091c\u093e\u0924\u093e \u0939\u0948\u0964<\/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=\"\u0915\u093e\u0930\u094d\u092f \u0915\u0940 \u0938\u0940\u092e\u093e\" 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=\"\u0915\u093e\u0930\u094d\u092f \u0915\u0940 \u0938\u0940\u092e\u093e\" class=\"alignnone size-full lazyload\" width=\"692\" height=\"565\" \/><\/noscript><\/center><\/p>\n<p style=\"text-align: justify;\">\u092f\u0926\u093f <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x<\/span><\/span> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x_0<\/span><\/span> \u0915\u0947 \u092a\u093e\u0938 \u0939\u0948, \u0924\u094b \u090f\u0915 \u0916\u0941\u0932\u0947 \u0905\u0902\u0924\u0930\u093e\u0932 \u0915\u093e \u0905\u0938\u094d\u0924\u093f\u0924\u094d\u0935 \u0939\u094b\u0917\u093e \u091c\u093f\u0938\u0915\u093e \u0930\u0947\u0921\u093f\u092f\u0938 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\delta<\/span><\/span> \u0914\u0930 \u0915\u0947\u0902\u0926\u094d\u0930 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x_0<\/span><\/span> \u0939\u0948, \u091c\u093f\u0938\u092e\u0947\u0902 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x<\/span><\/span> \u0939\u094b\u0917\u093e\u0964 \u0939\u092e \u0907\u0938\u0947 \u0924\u0940\u0928 \u0905\u0932\u0917-\u0905\u0932\u0917 \u0930\u0942\u092a\u094b\u0902 \u092e\u0947\u0902 \u0926\u0930\u094d\u0936\u093e \u0938\u0915\u0924\u0947 \u0939\u0948\u0902:<\/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;\">\u092f\u093e \u092b\u093f\u0930 <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>\u0939\u092e\u093e\u0930\u0947 \u0938\u0902\u0926\u0930\u094d\u092d \u092e\u0947\u0902, \u092f\u0947 \u0924\u0940\u0928\u094b\u0902 \u0924\u0930\u0940\u0915\u0947 \u090f\u0915 \u0939\u0940 \u092c\u093e\u0924 \u0915\u094b \u0915\u0939\u0924\u0947 \u0939\u0948\u0902; \u0939\u093e\u0932\u093e\u0902\u0915\u093f \u0905\u0902\u0924\u093f\u092e \u0924\u0930\u0940\u0915\u093e, \u091c\u093f\u0938\u0947 \u00ab\u0915\u0947\u0902\u0926\u094d\u0930 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x_0<\/span><\/span> \u0914\u0930 \u0930\u0947\u0921\u093f\u092f\u0938 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\delta<\/span><\/span> \u0915\u0940 \u0916\u0941\u0932\u0940 \u0917\u0947\u0902\u0926 \u092e\u0947\u0902 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x<\/span><\/span> \u0915\u0947 \u0930\u0942\u092a \u092e\u0947\u0902 \u092a\u0922\u093c\u093e \u091c\u093e\u0924\u093e \u0939\u0948, \u090f\u0915 <strong>\u091f\u094b\u092a\u094b\u0932\u0949\u091c\u0940 \u092a\u093e\u0920\u094d\u092f\u0915\u094d\u0930\u092e<\/strong> \u0915\u0947 \u0932\u093f\u090f \u0905\u0927\u093f\u0915 \u0909\u092a\u092f\u0941\u0915\u094d\u0924 \u0939\u094b \u0938\u0915\u0924\u093e \u0939\u0948, \u091c\u0939\u093e\u0901 \u0907\u0938 \u00ab\u0928\u093f\u0915\u091f\u0924\u093e \u0915\u0947 \u0935\u093f\u0937\u092f\u00bb \u092a\u0930 \u0905\u0927\u093f\u0915 \u0917\u0939\u0930\u093e\u0908 \u0938\u0947 \u0935\u093f\u091a\u093e\u0930 \u0915\u093f\u092f\u093e \u091c\u093e\u090f\u0917\u093e\u0964<\/em><\/p>\n<p style=\"text-align: justify;\">\u092f\u0926\u093f \u0910\u0938\u093e \u0939\u094b\u0924\u093e \u0939\u0948, \u0924\u094b \u0939\u092e \u0926\u0947\u0916\u0947\u0902\u0917\u0947 \u0915\u093f \u0915\u0947\u0902\u0926\u094d\u0930 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">l<\/span><\/span> \u0914\u0930 \u0930\u0947\u0921\u093f\u092f\u0938 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\epsilon<\/span><\/span> \u0915\u0947 \u0938\u093e\u0925 \u090f\u0915 \u0905\u0928\u094d\u092f \u0916\u0941\u0932\u0947 \u0905\u0902\u0924\u0930\u093e\u0932 \u0915\u093e \u0905\u0938\u094d\u0924\u093f\u0924\u094d\u0935 \u0939\u094b\u0917\u093e \u091c\u093f\u0938\u092e\u0947\u0902 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f(x)<\/span><\/span> \u0936\u093e\u092e\u093f\u0932 \u0939\u094b\u0917\u093e, \u0905\u0930\u094d\u0925\u093e\u0924: <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=\"\u0915\u093e\u0930\u094d\u092f \u0915\u0940 \u0938\u0940\u092e\u093e\" 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=\"\u0915\u093e\u0930\u094d\u092f \u0915\u0940 \u0938\u0940\u092e\u093e\" class=\"alignnone size-full lazyload\" width=\"625\" height=\"549\" \/><\/noscript><\/center><\/p>\n<p style=\"text-align: justify;\">\u092f\u0939\u0940\u0902 \u0938\u0947 \u0917\u0923\u093f\u0924\u0940\u092f \u0938\u0940\u092e\u093e \u0915\u0940 \u0905\u0935\u0927\u093e\u0930\u0923\u093e \u0915\u093e \u092e\u0942\u0932 \u0935\u093f\u091a\u093e\u0930 \u0909\u092d\u0930\u0924\u093e \u0939\u0948, \u0905\u0930\u094d\u0925\u093e\u0924 \u092f\u0939 \u0924\u092c \u092e\u094c\u091c\u0942\u0926 \u0939\u094b\u0917\u0940 \u091c\u092c: \u092f\u0926\u093f <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0 \\lt|x-x_0|\\lt \\delta<\/span><\/span>, \u0924\u094b <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">|f(x)-l|\\lt \\epsilon<\/span><\/span> \u0939\u094b\u0917\u093e; \u0914\u0930 \u092f\u0939 \u092e\u093e\u0928 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">l<\/span><\/span> \u0909\u0938 \u0915\u093e\u0930\u094d\u092f \u0915\u0940 \u0938\u0940\u092e\u093e \u0939\u094b\u0917\u0940 \u091c\u092c <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x<\/span><\/span> \u091c\u093f\u0924\u0928\u093e \u091a\u093e\u0939\u0947\u0902 \u0909\u0924\u0928\u093e <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x_0<\/span><\/span> \u0915\u0947 \u0915\u0930\u0940\u092c \u091c\u093e\u0924\u093e \u0939\u0948\u0964<\/p>\n<p><a name=\"3\"><\/a><\/p>\n<h2>\u0938\u0940\u092e\u093e \u0915\u0940 \u0914\u092a\u091a\u093e\u0930\u093f\u0915 \u092a\u0930\u093f\u092d\u093e\u0937\u093e<\/h2>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=FEPfoAfPsFY&amp;t=689s\" rel=\"noopener\" target=\"_blank\"><strong>\u091c\u094b \u0938\u0939\u091c \u0914\u0930 \u0917\u094d\u0930\u093e\u092b\u093f\u0915 \u0905\u0935\u0927\u093e\u0930\u0923\u093e \u092a\u094d\u0930\u0938\u094d\u0924\u0941\u0924 \u0915\u0940 \u0917\u0908 \u0939\u0948, \u0909\u0938\u0915\u0947 \u0906\u0927\u093e\u0930 \u092a\u0930 \u0939\u092e \u0938\u0940\u092e\u093e \u0915\u0940 \u0914\u092a\u091a\u093e\u0930\u093f\u0915 \u092a\u0930\u093f\u092d\u093e\u0937\u093e \u092a\u0930 \u0935\u093f\u091a\u093e\u0930 \u0915\u0930 \u0938\u0915\u0924\u0947 \u0939\u0948\u0902\u0964<\/strong> <\/a> \u0939\u092e \u0915\u0939\u0924\u0947 \u0939\u0948\u0902 \u0915\u093f \u0938\u0940\u092e\u093e \u0924\u092c \u092e\u094c\u091c\u0942\u0926 \u0939\u0948 \u091c\u092c, \u091a\u093e\u0939\u0947 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\epsilon<\/span><\/span> (\u092f\u093e\u0928\u0940 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f(x)<\/span><\/span> \u0914\u0930 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">l<\/span><\/span> \u0915\u0947 \u092c\u0940\u091a \u0915\u0940 \u0926\u0942\u0930\u0940) \u091c\u094b \u092d\u0940 \u0939\u094b, \u0939\u092e\u0947\u0936\u093e \u090f\u0915 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\delta<\/span><\/span> \u0939\u094b\u0917\u093e \u0924\u093e\u0915\u093f \u092f\u0926\u093f <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0 \\lt|x-x_0|\\lt \\delta<\/span><\/span> \u0939\u0948, \u0924\u094b <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">|f(x) - l|\\lt \\epsilon.<\/span><\/span> \u092f\u0939 \u0935\u093f\u091a\u093e\u0930 \u0936\u0941\u0930\u0941\u0906\u0924 \u092e\u0947\u0902 \u092a\u0915\u0921\u093c\u0928\u093e \u092e\u0941\u0936\u094d\u0915\u093f\u0932 \u0939\u094b \u0938\u0915\u0924\u093e \u0939\u0948 \u0914\u0930 \u092f\u0939 \u0926\u0941\u0928\u093f\u092f\u093e \u092d\u0930 \u0915\u0947 \u0905\u0927\u093f\u0915\u093e\u0902\u0936 \u0917\u0923\u0928\u093e \u091b\u093e\u0924\u094d\u0930\u094b\u0902 \u0915\u0947 \u0932\u093f\u090f \u0906\u0901\u0938\u0942 \u0915\u093e \u0915\u093e\u0930\u0923 \u0939\u094b\u0924\u093e \u0939\u0948, \u0932\u0947\u0915\u093f\u0928 \u0907\u0938\u0947 \u0928\u093f\u092e\u094d\u0928\u0932\u093f\u0916\u093f\u0924 \u0905\u092d\u093f\u0935\u094d\u092f\u0915\u094d\u0924\u093f \u092e\u0947\u0902 \u0938\u0902\u0915\u094d\u0937\u0947\u092a\u093f\u0924 \u0915\u093f\u092f\u093e \u091c\u093e \u0938\u0915\u0924\u093e \u0939\u0948:<\/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>\u0938\u0940\u092e\u093e\u0913\u0902 \u0915\u0947 \u0917\u0941\u0923<\/h2>\n<p style=\"text-align: justify;\">\u0938\u0940\u092e\u093e \u0915\u0940 \u0914\u092a\u091a\u093e\u0930\u093f\u0915 \u092a\u0930\u093f\u092d\u093e\u0937\u093e \u0915\u093e \u092e\u0939\u0924\u094d\u0935 \u092f\u0939 \u0939\u0948 \u0915\u093f \u0905\u092c, \u0907\u0938\u0915\u0947 \u0906\u0927\u093e\u0930 \u092a\u0930, \u0939\u092e \u0907\u0938\u0915\u0947 \u0917\u0941\u0923\u094b\u0902 \u0915\u094b \u0938\u093f\u0926\u094d\u0927 \u0915\u0930 \u0938\u0915\u0924\u0947 \u0939\u0948\u0902, \u091a\u093e\u0939\u0947 \u0935\u0947 \u0938\u0939\u091c \u0939\u094b\u0902 \u092f\u093e \u0905\u0928\u094d\u092f \u0917\u0941\u0923 \u091c\u094b \u0907\u0924\u0928\u0947 \u0938\u0939\u091c \u0928\u0939\u0940\u0902 \u0939\u0948\u0902\u0964<\/p>\n<p style=\"text-align: justify;\">\u0906\u0917\u0947 \u092c\u0922\u093c\u0928\u0947 \u0938\u0947 \u092a\u0939\u0932\u0947, \u091c\u092c\u0915\u093f \u092f\u0939 \u0938\u0916\u094d\u0924\u0940 \u0938\u0947 \u0906\u0935\u0936\u094d\u092f\u0915 \u0928\u0939\u0940\u0902 \u0939\u0948, \u092f\u0939 \u0905\u0924\u094d\u092f\u0927\u093f\u0915 \u0905\u0928\u0941\u0936\u0902\u0938\u093e \u0915\u0940 \u091c\u093e\u0924\u0940 \u0939\u0948 \u0915\u093f \u0906\u092a <a href=\"http:\/\/toposuranos.com\/material\/es\/category\/matematica\/logica-matematica\/logica-proposicional\/\" rel=\"noopener\" target=\"_blank\"><strong>\u0917\u0923\u093f\u0924\u0940\u092f \u0924\u0930\u094d\u0915<\/strong><\/a> \u0915\u0940 \u0915\u0941\u091b \u0905\u0935\u0927\u093e\u0930\u0923\u093e\u0913\u0902 \u0915\u0940 \u0938\u092e\u0940\u0915\u094d\u0937\u093e \u0915\u0930\u0947\u0902 \u0924\u093e\u0915\u093f \u0906\u092a \u0906\u0917\u093e\u092e\u0940 \u092a\u094d\u0930\u0926\u0930\u094d\u0936\u0928\u094b\u0902 \u0915\u094b \u0905\u0927\u093f\u0915 \u0906\u0938\u093e\u0928\u0940 \u0938\u0947 \u0938\u092e\u091d \u0938\u0915\u0947\u0902\u0964<\/p>\n<p><a name=\"5\"><\/a><\/p>\n<h3>\u092f\u0926\u093f \u0938\u0940\u092e\u093e \u092e\u094c\u091c\u0942\u0926 \u0939\u0948, \u0924\u094b \u092f\u0939 \u0905\u0926\u094d\u0935\u093f\u0924\u0940\u092f \u0939\u0948<\/h3>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=FEPfoAfPsFY&amp;t=904s\" rel=\"noopener\" target=\"_blank\"><strong>\u0907\u0938 \u0917\u0941\u0923 \u0915\u094b \u0938\u093f\u0926\u094d\u0927 \u0915\u0930\u0928\u0947 \u0915\u0947 \u0932\u093f\u090f, \u0939\u092e \u092a\u094d\u0930\u0924\u093f\u0935\u093e\u0926 \u0915\u0947 \u0926\u094d\u0935\u093e\u0930\u093e \u0938\u093f\u0926\u094d\u0927\u093f \u0915\u0940 \u0924\u0915\u0928\u0940\u0915 \u0915\u093e \u0909\u092a\u092f\u094b\u0917 \u0915\u0930\u0947\u0902\u0917\u0947\u0964<\/strong><\/a> \u0939\u092e \u0928\u093f\u092e\u094d\u0928\u0932\u093f\u0916\u093f\u0924 \u092a\u094d\u0930\u0938\u094d\u0925\u093e\u092a\u0928\u093e\u0913\u0902 \u0915\u093e \u0938\u092e\u0942\u0939 \u092a\u0930\u093f\u092d\u093e\u0937\u093f\u0924 \u0915\u0930\u0924\u0947 \u0939\u0941\u090f \u0936\u0941\u0930\u0941\u0906\u0924 \u0915\u0930\u0947\u0902\u0917\u0947:<\/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;\">\u0907\u0938\u0915\u0947 \u0906\u0927\u093e\u0930 \u092a\u0930 \u0939\u092e \u0928\u093f\u092e\u094d\u0928\u0932\u093f\u0916\u093f\u0924 \u0914\u092a\u091a\u093e\u0930\u093f\u0915 \u092a\u094d\u0930\u092e\u093e\u0923 \u092c\u0928\u093e \u0938\u0915\u0924\u0947 \u0939\u0948\u0902:<\/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>\u092e\u093e\u0928\u094d\u092f\u0924\u093e<\/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>; \u092e\u093e\u0928\u094d\u092f\u0924\u093e<\/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>; \u092e\u093e\u0928\u094d\u092f\u0924\u093e<\/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>\u0905\u0902\u0924\u0930\u094d\u0917\u094d\u0930\u0939\u0923<\/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>\u090f\u0915\u0930\u0942\u092a\u0924\u093e<\/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>; \u0915\u094d\u092f\u094b\u0902\u0915\u093f <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>; \u092a\u094d\u0930\u092f\u094b\u0917(5,6)<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td style=\" text-align: left;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\mathcal{H}\\cup\\{L\\lt L^\\prime\\}\\vdash (\\exists \\delta\\gt 0) (0 \\lt|x-x_0|\\lt\\delta \\rightarrow [<\/span><\/span> <span style=\"background-color: #a0ffa0; color:#000000\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">( 2 |f(x) - L |\\lt L - L^\\prime )<\/span><\/span><\/span> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\wedge<\/span><\/span> <span style=\"background-color: #a0a0ff; color:#000000\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">( 2|f(x) - L^\\prime |\\lt L - L^\\prime)<\/span><\/span><\/span> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\"> ])<\/span><\/span><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td style=\" text-align: left;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\mathcal{H}\\cup\\{L\\lt L^\\prime\\}\\vdash (\\exists \\delta\\gt 0) (0 \\lt|x-x_0|\\lt\\delta \\rightarrow [<\/span><\/span> <span style=\"background-color: #a0ffa0; color:#000000\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">( -L + L^\\prime \\lt 2 (f(x) - L )\\lt L - L^\\prime )<\/span><\/span><\/span> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\wedge<\/span><\/span> <span style=\"background-color: #a0a0ff; color:#000000\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">( -L + L^\\prime \\lt 2(f(x) - L^\\prime )\\lt L - L^\\prime)<\/span><\/span><\/span> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\"> ])<\/span><\/span><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td style=\" text-align: left;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\mathcal{H}\\cup\\{L\\lt L^\\prime\\}\\vdash (\\exists \\delta\\gt 0) (0 \\lt|x-x_0|\\lt\\delta \\rightarrow [<\/span><\/span> <span style=\"background-color: #a0ffa0; color:#000000\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">( -L + L^\\prime \\lt 2f(x) - 2L \\lt L - L^\\prime )<\/span><\/span><\/span> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\wedge<\/span><\/span> <span style=\"background-color: #a0a0ff; color:#000000\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">( -L + L^\\prime \\lt 2f(x) - 2L^\\prime \\lt L - 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>; De(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>; \u0909\u0938\u0940 \u0935\u093f\u0927\u093f (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>-int(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>; Def(10)<\/td>\n<\/tr>\n<tr>\n<td style=\" text-align: right;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(12)<\/span><\/span><\/td>\n<td style=\" text-align: left;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\mathcal{H}\\vdash \\bot <\/span><\/span>; <strong>MP<\/strong>(3,11)<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td style=\" text-align: left;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\left\\{\\lim_{x\\to x_0}f(x) = L, \\lim_{x\\to x_0}f(x) = L^\\prime, L\\neq L^\\prime\\right\\} \\vdash \\bot <\/span><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\" text-align: right;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(13)<\/span><\/span><\/td>\n<td style=\" text-align: left;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\left\\{\\lim_{x\\to x_0}f(x) = L, \\lim_{x\\to x_0}f(x) = L^\\prime \\right\\} \\vdash \\neg(L\\neq L^\\prime) <\/span><\/span>; <strong>\u0935\u093f\u0930\u094b\u0927\u093e\u092d\u093e\u0938<\/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;\">\u0907\u0938 \u092a\u094d\u0930\u092e\u093e\u0923 \u0938\u0947 \u0939\u092e \u092a\u094d\u0930\u093e\u092a\u094d\u0924 \u0915\u0930\u0924\u0947 \u0939\u0948\u0902 \u0915\u093f, \u092f\u0926\u093f \u0926\u094b \u0938\u0940\u092e\u093e\u090f\u0902 \u092e\u094c\u091c\u0942\u0926 \u0939\u0948\u0902, \u0924\u094b \u0935\u0947 \u0938\u092e\u093e\u0928 \u0939\u0948\u0902 \u0914\u0930 \u0907\u0938\u0932\u093f\u090f, \u0938\u0940\u092e\u093e \u0905\u0926\u094d\u0935\u093f\u0924\u0940\u092f \u0939\u0948\u0964<\/p>\n<p><a name=\"6\"><\/a><\/p>\n<h3>\u0938\u0940\u092e\u093e\u0913\u0902 \u0915\u093e \u092c\u0940\u091c\u0917\u0923\u093f\u0924<\/h3>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=FEPfoAfPsFY&amp;t=2011s\" rel=\"noopener\" target=\"_blank\"><strong>\u0905\u092c \u0924\u0915 \u0926\u0947\u0916\u0947 \u0917\u090f \u0924\u0925\u094d\u092f\u094b\u0902 \u0915\u0947 \u0906\u0927\u093e\u0930 \u092a\u0930 \u0939\u092e \u0938\u0940\u092e\u093e\u0913\u0902 \u0915\u0947 \u0917\u0923\u093f\u0924\u0940\u092f \u0935\u093f\u091a\u093e\u0930 \u0915\u093e \u092e\u0942\u0932\u092d\u0942\u0924 \u0905\u0927\u094d\u092f\u092f\u0928 \u0915\u0930\u0924\u0947 \u0939\u0948\u0902\u0964<\/strong><\/a> \u0932\u0947\u0915\u093f\u0928 \u092f\u0939 \u0905\u092d\u0940 \u092d\u0940 \u092a\u0930\u094d\u092f\u093e\u092a\u094d\u0924 \u0928\u0939\u0940\u0902 \u0939\u0948 \u0915\u093f \u0939\u092e \u0938\u0940\u092e\u093e\u0913\u0902 \u0915\u0947 \u0938\u093e\u0925 \u0917\u0923\u0928\u093e \u0915\u0930 \u0938\u0915\u0947\u0902\u0964 \u0915\u0947\u0935\u0932 \u090f\u0915 \u092a\u093e\u0917\u0932 \u0939\u0940 \u0907\u0938\u0915\u093e \u0909\u092a\u092f\u094b\u0917 \u0907\u0938 \u0909\u0926\u094d\u0926\u0947\u0936\u094d\u092f \u0915\u0947 \u0932\u093f\u090f \u0915\u0930\u0947\u0917\u093e\u0964 \u0907\u0938 \u0938\u092e\u0938\u094d\u092f\u093e \u0915\u094b \u0939\u0932 \u0915\u0930\u0928\u0947 \u0915\u0947 \u0932\u093f\u090f, \u0905\u092c \u0939\u092e \u0909\u0928 \u0924\u0915\u0928\u0940\u0915\u094b\u0902 \u092a\u0930 \u0915\u093e\u092e \u0915\u0930\u0947\u0902\u0917\u0947 \u091c\u094b \u0939\u092e\u0947\u0902 \u0915\u0941\u091b \u0938\u0940\u092e\u093e\u0913\u0902 \u0915\u0940 \u0917\u0923\u0928\u093e \u0936\u0941\u0930\u0942 \u0915\u0930\u0928\u0947 \u092e\u0947\u0902 \u092e\u0926\u0926 \u0915\u0930\u0947\u0902\u0917\u0940\u0964<\/p>\n<p style=\"text-align: justify;\">\u092e\u093e\u0928 \u0932\u0947\u0902 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x_0, \\alpha, \\beta, L, M \\in \\mathbb{R},<\/span><\/span> \u0914\u0930 f \u0924\u0925\u093e g \u0935\u093e\u0938\u094d\u0924\u0935\u093f\u0915 \u092b\u0932\u0928 \u0939\u094b\u0902 \u091c\u093f\u0928\u0915\u0947 \u0932\u093f\u090f:<\/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;\">\u0924\u092c \u0928\u093f\u092e\u094d\u0928\u0932\u093f\u0916\u093f\u0924 \u0917\u0941\u0923 \u0938\u0924\u094d\u092f \u0939\u094b\u0924\u0947 \u0939\u0948\u0902:<\/p>\n<h4>\u092b\u0932\u0928\u094b\u0902 \u0915\u0947 \u092f\u094b\u0917 \u0914\u0930 \u0905\u0902\u0924\u0930 \u0915\u0940 \u0938\u0940\u092e\u093e<\/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>\u0938\u093f\u0926\u094d\u0927\u093f:<\/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>\u092a\u094d\u0930\u0938\u094d\u0925\u093e\u092a\u0928\u094b\u0902 \u0915\u0947 \u0938\u092e\u0942\u0939 \u092a\u0930 \u0935\u093f\u091a\u093e\u0930 \u0915\u0930\u0947\u0902<\/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>, \u0924\u092c \u0907\u0938\u0915\u0947 \u0906\u0927\u093e\u0930 \u092a\u0930 \u0939\u092e \u0928\u093f\u092e\u094d\u0928\u0932\u093f\u0916\u093f\u0924 \u0924\u0930\u094d\u0915 \u0915\u0930 \u0938\u0915\u0924\u0947 \u0939\u0948\u0902:<\/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>; \u092e\u093e\u0928\u094d\u092f\u0924\u093e<\/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>; \u092a\u0930\u093f\u092d\u093e\u0937\u093e<\/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>; (1,2) \u0938\u0947<\/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>; \u092e\u093e\u0928\u094d\u092f\u0924\u093e<\/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>; (3) \u0915\u0947 \u0938\u092e\u093e\u0928<\/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>; (3,5) \u0938\u0947<\/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>; \u0924\u094d\u0930\u093f\u092d\u0941\u091c \u0905\u0938\u092e\u093e\u0928\u0924\u093e: <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>; (6,7) \u0938\u0947<\/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>; \u092a\u0930\u093f\u092d\u093e\u0937\u093e<\/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>; (8,9) \u0938\u0947<\/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>; \u092a\u0930\u093f\u092d\u093e\u0937\u093e<\/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>; (10) \u0938\u0947 \u0935\u094d\u092f\u0941\u0924\u094d\u092a\u0924\u094d\u0924\u093f<\/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>; (11,12) \u0938\u0947<\/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>; (10,13) \u0938\u0947<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h4>\u092b\u0902\u0915\u094d\u0936\u0928\u094b\u0902 \u0915\u0947 \u0917\u0941\u0923\u0928\u092b\u0932 \u0915\u0940 \u0938\u0940\u092e\u093e<\/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>\u092f\u0939 \u0938\u093f\u0926\u094d\u0927\u093f \u092a\u093f\u091b\u0932\u0947 \u0935\u093e\u0932\u0940 \u0938\u0947 \u0925\u094b\u0921\u093c\u0940 \u0905\u0927\u093f\u0915 \u0915\u0920\u093f\u0928 \u0939\u0948,<\/strong><\/a> \u0932\u0947\u0915\u093f\u0928 \u092f\u0939 \u0915\u0941\u091b \u091f\u094d\u0930\u093f\u0915\u094d\u0938 \u0915\u0940 \u092e\u0926\u0926 \u0938\u0947 \u0939\u0932 \u0915\u0940 \u091c\u093e \u0938\u0915\u0924\u0940 \u0939\u0948\u0964 \u092a\u093f\u091b\u0932\u0947 \u0938\u093f\u0926\u094d\u0927\u093e\u0902\u0924 \u0915\u0947 \u0938\u092e\u093e\u0928 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathcal{H}<\/span><\/span> \u0915\u0947 \u0938\u093e\u0925 \u0939\u092e \u0928\u093f\u092e\u094d\u0928\u0932\u093f\u0916\u093f\u0924 \u0924\u0930\u094d\u0915 \u0915\u0930 \u0938\u0915\u0924\u0947 \u0939\u0948\u0902:<\/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>; \u092a\u0930\u093f\u092d\u093e\u0937\u093e<\/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>; \u092e\u093e\u0928\u094d\u092f\u0924\u093e<\/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>; (1) \u0915\u093e \u0909\u092a\u092f\u094b\u0917<\/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>; \u092a\u0930\u093f\u092d\u093e\u0937\u093e<\/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>; \u092e\u093e\u0928\u094d\u092f\u0924\u093e<\/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>; (3) \u0915\u093e \u0909\u092a\u092f\u094b\u0917<\/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>; \u0924\u094d\u0930\u093f\u0915\u094b\u0923\u0940\u092f \u0905\u0938\u092e\u093e\u0928\u0924\u093e + <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\overline{\\epsilon}<\/span><\/span> \u0915\u093e \u0935\u093f\u0936\u0947\u0937 \u092e\u093e\u092e\u0932\u093e<\/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>; (5) \u0938\u0947<\/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>; \u0924\u094d\u0930\u093f\u0915\u094b\u0923\u0940\u092f \u0905\u0938\u092e\u093e\u0928\u0924\u093e + <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\overline{\\overline{\\epsilon}}<\/span><\/span> \u0915\u093e \u0935\u093f\u0936\u0947\u0937 \u092e\u093e\u092e\u0932\u093e<\/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>; (7) \u0938\u0947<\/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>; \u0936\u0942\u0928\u094d\u092f \u0915\u094b \u091c\u094b\u0921\u093c\u0947\u0902<\/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>; \u0917\u0941\u0923\u0928<\/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>; \u0924\u094d\u0930\u093f\u0915\u094b\u0923\u0940\u092f \u0905\u0938\u092e\u093e\u0928\u0924\u093e(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>; (5,6,10) \u0938\u0947<\/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>; (11) \u0938\u0947<\/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>; (1,3,12) \u0938\u0947<\/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>; (11,13) \u0938\u0947<\/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>; (1,2,4,14) \u0938\u0947<\/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>\u0938\u094d\u0925\u093f\u0930 \u092b\u0932\u0928 \u0915\u0940 \u0938\u0940\u092e\u093e<\/h4>\n<p style=\"text-align: justify; \"><a href=\"https:\/\/www.youtube.com\/watch?v=FEPfoAfPsFY&amp;t=3450s\" rel=\"noopener\" target=\"_blank\"><strong>\u0938\u094d\u0925\u093f\u0930 \u092b\u0932\u0928 \u0915\u0940 \u0938\u0940\u092e\u093e<\/strong><\/a> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f(x)=c<\/span><\/span>, \u0938\u094d\u0925\u093f\u0930\u093e\u0902\u0915 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">c<\/span><\/span> \u0939\u0948\u0964 \u0905\u0930\u094d\u0925\u093e\u0924\u094d<\/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>\u092a\u094d\u0930\u092e\u093e\u0923<\/strong><\/p>\n<p style=\"text-align: justify; \">\u0907\u0938\u0915\u093e \u092a\u094d\u0930\u092e\u093e\u0923 \u0935\u093e\u0938\u094d\u0924\u0935 \u092e\u0947\u0902 \u0938\u0930\u0932 \u0939\u0948, \u0915\u094d\u092f\u094b\u0902\u0915\u093f \u092f\u0939 \u090f\u0915 \u0924\u093e\u0924\u094d\u0924\u094d\u0935\u093f\u0915 \u0938\u0924\u094d\u092f \u0939\u0948\u0964 \u092a\u0939\u0932\u0947 \u0938\u0947 \u0939\u0940 \u091c\u094d\u091e\u093e\u0924 \u0939\u0948 \u0915\u093f:<\/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; \">\u0932\u0947\u0915\u093f\u0928 \u092f\u0939 \u0939\u094b\u0924\u093e \u0939\u0948 \u0915\u093f <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0=|c-c|\\lt \\epsilon<\/span><\/span> \u090f\u0915 \u0924\u093e\u0924\u094d\u0924\u094d\u0935\u093f\u0915 \u0938\u0924\u094d\u092f \u0939\u0948 \u0939\u0930 \u0938\u0915\u093e\u0930\u093e\u0924\u094d\u092e\u0915 \u090f\u092a\u094d\u092a\u094d\u0938\u093f\u0932\u0949\u0928 \u0915\u0947 \u0932\u093f\u090f, \u0907\u0938\u0932\u093f\u090f \u092f\u0939 \u0928\u093f\u0939\u093f\u0924\u093e\u0930\u094d\u0925 \u092d\u0940 \u0924\u093e\u0924\u094d\u0924\u094d\u0935\u093f\u0915 \u0938\u0924\u094d\u092f \u0939\u0948 \u0914\u0930 \u092a\u0930\u093f\u0923\u093e\u092e\u0938\u094d\u0935\u0930\u0942\u092a, <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to x_0}c = c <\/span><\/span> \u092d\u0940 \u090f\u0915 \u0924\u093e\u0924\u094d\u0924\u094d\u0935\u093f\u0915 \u0938\u0924\u094d\u092f \u0939\u0948\u0964<\/p>\n<h4>\u092b\u0932\u0928\u094b\u0902 \u0915\u0947 \u0905\u0928\u0941\u092a\u093e\u0924 \u0915\u0940 \u0938\u0940\u092e\u093e<\/h4>\n<p style=\"text-align: justify; \"><a href=\"https:\/\/www.youtube.com\/watch?v=FEPfoAfPsFY&amp;t=3563s\" rel=\"noopener\" target=\"_blank\"><strong>\u0905\u092c \u0939\u092e \u0926\u094b \u092b\u0932\u0928\u094b\u0902 \u0915\u0947 \u0905\u0928\u0941\u092a\u093e\u0924 \u0915\u0940 \u0938\u0940\u092e\u093e \u0915\u0947 \u0928\u093f\u092f\u092e \u0915\u094b \u0938\u093e\u092c\u093f\u0924 \u0915\u0930\u0928\u0947 \u0915\u0947 \u0932\u093f\u090f \u0924\u0948\u092f\u093e\u0930 \u0939\u0948\u0902\u0964<\/strong><\/a> \u092f\u0939 \u0939\u0948<\/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; \">\u091c\u0939\u093e\u0901, \u092a\u093f\u091b\u0932\u0940 \u0917\u0941\u0923\u094b\u0902 \u0915\u0940 \u0924\u0930\u0939, \u0939\u092e \u092f\u0939 \u092e\u093e\u0928\u0924\u0947 \u0939\u0948\u0902 \u0915\u093f \u0928\u093f\u092e\u094d\u0928\u0932\u093f\u0916\u093f\u0924 \u092a\u0942\u0930\u094d\u0935\u0927\u093e\u0930\u093e\u090f\u0901 \u092a\u0942\u0930\u0940 \u0939\u094b\u0924\u0940 \u0939\u0948\u0902:<\/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>\u092a\u094d\u0930\u092e\u093e\u0923<\/strong><\/p>\n<p style=\"text-align: justify; \">\u0938\u094c\u092d\u093e\u0917\u094d\u092f \u0938\u0947, \u0939\u092e\u0947\u0902 \u092a\u0939\u0932\u0947 \u0915\u093f\u090f \u0917\u090f \u092a\u094d\u0930\u092e\u093e\u0923\u094b\u0902 \u091c\u0948\u0938\u0947 \u0914\u0930 \u092a\u094d\u0930\u092e\u093e\u0923 \u0928\u0939\u0940\u0902 \u0915\u0930\u0928\u0947 \u0939\u094b\u0902\u0917\u0947, \u0915\u094d\u092f\u094b\u0902\u0915\u093f \u0905\u092c \u0939\u092e \u0938\u0940\u0927\u0947 \u0909\u0928 \u092a\u0930\u093f\u0923\u093e\u092e\u094b\u0902 \u0915\u093e \u0909\u092a\u092f\u094b\u0917 \u0915\u0930 \u0938\u0915\u0924\u0947 \u0939\u0948\u0902\u0964 \u0932\u0947\u0915\u093f\u0928 \u0907\u0938\u0938\u0947 \u092a\u0939\u0932\u0947, \u0939\u092e \u0938\u092c\u0938\u0947 \u092a\u0939\u0932\u0947 \u092f\u0939 \u0938\u093e\u092c\u093f\u0924 \u0915\u0930\u0947\u0902\u0917\u0947 \u0915\u093f<\/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; \">\u092f\u0939 \u0938\u093e\u092c\u093f\u0924 \u0915\u0930\u0928\u0947 \u0915\u0947 \u0932\u093f\u090f \u0909\u0924\u094d\u092a\u093e\u0926 \u0915\u0940 \u0938\u0940\u092e\u093e \u0914\u0930 \u0938\u094d\u0925\u093f\u0930 \u092b\u0932\u0928 \u0915\u0940 \u0938\u0940\u092e\u093e \u0915\u0947 \u0928\u093f\u092f\u092e \u0915\u093e \u0909\u092a\u092f\u094b\u0917 \u0915\u0930\u0928\u093e \u092a\u0930\u094d\u092f\u093e\u092a\u094d\u0924 \u0939\u0948, \u092c\u0938 \u0939\u092e\u0947\u0902 \u0907\u0938 \u092c\u093e\u0924 \u0915\u093e \u0927\u094d\u092f\u093e\u0928 \u0930\u0916\u0928\u093e \u091a\u093e\u0939\u093f\u090f \u0915\u093f <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">g(x)<\/span><\/span> \u0936\u0942\u0928\u094d\u092f \u0928\u0939\u0940\u0902 \u0939\u094b\u0928\u0940 \u091a\u093e\u0939\u093f\u090f:<\/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; \">\u0907\u0938\u0932\u093f\u090f: <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; \">\u0905\u0902\u0924 \u092e\u0947\u0902, \u0909\u0924\u094d\u092a\u093e\u0926 \u0915\u0940 \u0938\u0940\u092e\u093e \u0915\u0947 \u0928\u093f\u092f\u092e \u0915\u0947 \u0905\u0928\u0941\u0938\u093e\u0930:<\/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; \">\u092f\u0939 \u092e\u093e\u0928 \u0924\u092c \u0938\u0939\u0940 \u0939\u094b\u0917\u093e \u091c\u092c <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">M<\/span><\/span> \u0936\u0942\u0928\u094d\u092f \u0928 \u0939\u094b\u0964<\/p>\n<h4>\u092a\u094d\u0930\u093e\u0915\u0943\u0924\u093f\u0915 \u0918\u093e\u0924\u093e\u0902\u0915 \u0915\u0940 \u0938\u0940\u092e\u093e<\/h4>\n<p style=\"text-align: justify; \"><a href=\"https:\/\/www.youtube.com\/watch?v=FEPfoAfPsFY&amp;t=3725s\" rel=\"noopener\" target=\"_blank\"><strong>\u092f\u0939 \u0917\u0941\u0923 \u0939\u092e\u0947\u0902 \u092c\u0924\u093e\u0924\u0940 \u0939\u0948 \u0915\u093f,<\/strong><\/a> \u092f\u0926\u093f <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x_0 \\to x_0}f(x) = L<\/span><\/span>, \u0924\u092c \u092f\u0939 \u092e\u093e\u0928\u094d\u092f \u0939\u094b\u0917\u093e \u0915\u093f <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>\u0964 \u0907\u0938\u0947 \u0939\u092e \u0917\u0923\u093f\u0924\u0940\u092f \u0928\u093f\u0926\u0930\u094d\u0936\u0928 \u0926\u094d\u0935\u093e\u0930\u093e \u0938\u093f\u0926\u094d\u0927 \u0915\u0930 \u0938\u0915\u0924\u0947 \u0939\u0948\u0902\u0964<\/p>\n<p style=\"text-align: justify; color: #000080;\"><strong>\u092a\u094d\u0930\u092e\u093e\u0923:<\/strong><\/p>\n<ul style=\"text-align: justify; \">\n<li><strong>\u092e\u093e\u092e\u0932\u093e <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">n=1<\/span><\/span>:<\/strong> (\u092a\u094d\u0930\u093e\u0930\u0902\u092d\u093f\u0915 \u091a\u0930\u0923)\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> \u092f\u0939 \u092a\u094d\u0930\u093e\u0930\u0902\u092d\u093f\u0915 \u091a\u0930\u0923 \u0915\u094b \u092a\u0942\u0930\u093e \u0915\u0930\u0924\u093e \u0939\u0948 \u2705<\/p>\n<\/li>\n<li><strong>\u092e\u093e\u092e\u0932\u093e <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">n=k<\/span><\/span>:<\/strong> (\u0928\u093f\u0926\u0930\u094d\u0936\u0928 \u091a\u0930\u0923)\n<p style=\"text-align: justify;\">\u092e\u093e\u0928\u0924\u0947 \u0939\u0941\u090f \u0915\u093f <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to x_0} [f(x)]^k = L^k <\/span><\/span> (\u0928\u093f\u0926\u0930\u094d\u0936\u0928 \u0915\u0940 \u092a\u0930\u093f\u0915\u0932\u094d\u092a\u0928\u093e) \u092e\u093e\u0928\u094d\u092f \u0939\u0948, \u0939\u092e \u0926\u0947\u0916\u0947\u0902\u0917\u0947 \u0915\u093f <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> \u092d\u0940 \u092e\u093e\u0928\u094d\u092f \u0939\u0948\u0964<\/p>\n<p style=\"text-align: justify;\">\u0907\u0938\u092e\u0947\u0902: <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>\u0964 \u092f\u0939 \u0905\u0902\u0924\u093f\u092e \u092a\u0930\u093f\u0923\u093e\u092e \u090a\u092a\u0930 \u0926\u093f\u090f \u0917\u090f \u0909\u0924\u094d\u092a\u093e\u0926 \u0915\u0947 \u0938\u0940\u092e\u093e \u0915\u0947 \u0928\u093f\u092f\u092e \u092a\u0930 \u0906\u0927\u093e\u0930\u093f\u0924 \u0939\u0948\u0964<\/p>\n<p style=\"text-align: justify;\">\u0907\u0938\u0915\u0947 \u092c\u093e\u0926, \u0928\u093f\u0926\u0930\u094d\u0936\u0928 \u0915\u0940 \u092a\u0930\u093f\u0915\u0932\u094d\u092a\u0928\u093e \u0938\u0947 \u092f\u0939 \u0939\u094b\u0917\u093e <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> \u092f\u0939 \u0928\u093f\u0926\u0930\u094d\u0936\u0928 \u091a\u0930\u0923 \u0915\u094b \u092a\u0942\u0930\u093e \u0915\u0930\u0924\u093e \u0939\u0948 \u2705<\/p>\n<\/li>\n<li>\u0907\u0938\u0932\u093f\u090f: <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>n-\u0935\u0940\u0902 \u091c\u0921\u093c \u0915\u0940 \u0938\u0940\u092e\u093e<\/h4>\n<p style=\"text-align: justify; \"><a href=\"https:\/\/www.youtube.com\/watch?v=FEPfoAfPsFY&amp;t=3912s\" rel=\"noopener\" target=\"_blank\"><strong>\u0918\u093e\u0924\u093e\u0902\u0915 \u0915\u0940 \u0924\u0930\u0939 \u0939\u0940 \u092f\u0939 \u092e\u093e\u0928\u094d\u092f \u0939\u094b\u0917\u093e \u0915\u093f<\/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>\u092a\u094d\u0930\u092e\u093e\u0923:<\/strong><\/p>\n<p style=\"text-align: justify; \">\u090a\u092a\u0930 \u0938\u093f\u0926\u094d\u0927 \u0915\u093f\u090f \u0917\u090f \u0918\u093e\u0924\u093e\u0902\u0915 \u0915\u0947 \u0928\u093f\u092f\u092e \u0915\u093e \u0909\u092a\u092f\u094b\u0917 \u0915\u0930\u0915\u0947 \u0939\u092e \u092a\u093e\u0924\u0947 \u0939\u0948\u0902 \u0915\u093f<\/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; \">\u0907\u0938\u0932\u093f\u090f: <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>\u092d\u093f\u0928\u094d\u0928\u093e\u0924\u094d\u092e\u0915 \u0918\u093e\u0924\u093e\u0902\u0915\u094b\u0902 \u0915\u0940 \u0938\u0940\u092e\u093e<\/h4>\n<p style=\"text-align: justify; \"><a href=\"https:\/\/www.youtube.com\/watch?v=FEPfoAfPsFY&amp;t=4007s\" rel=\"noopener\" target=\"_blank\"><strong>\u092a\u093f\u091b\u0932\u0947 \u0926\u094b \u092a\u094d\u0930\u092e\u093e\u0923\u094b\u0902 \u0915\u0947 \u0938\u0902\u092f\u094b\u091c\u0928 \u0938\u0947<\/strong><\/a> \u0939\u092e \u0905\u092a\u0928\u0947 \u0905\u0902\u0924\u093f\u092e \u092a\u094d\u0930\u092e\u093e\u0923 \u092a\u0930 \u092a\u0939\u0941\u0901\u091a\u0924\u0947 \u0939\u0948\u0902, \u092f\u0939 \u0939\u0948: <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>, \u091c\u094b \u0909\u0924\u094d\u092a\u093e\u0926 \u0915\u0947 \u0928\u093f\u092f\u092e \u0915\u0947 \u0906\u0927\u093e\u0930 \u092a\u0930 \u0938\u093f\u0926\u094d\u0927 \u0939\u094b\u0924\u093e \u0939\u0948 \u0915\u094d\u092f\u094b\u0902\u0915\u093f <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle [f(x)]^{\\frac{p}{q}} =[\\sqrt[q]{f(x)}]^p <\/span><\/span> \u0914\u0930 <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>\u0938\u0940\u092e\u093e <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>\u0907\u0938 \u092a\u094d\u0930\u092e\u093e\u0923 \u0915\u0947 \u0938\u093e\u0925 \u0939\u092e \u0907\u0938 \u092a\u094d\u0930\u092e\u093e\u0923 \u0936\u0943\u0902\u0916\u0932\u093e \u0915\u094b \u0938\u092e\u093e\u092a\u094d\u0924 \u0915\u0930\u0924\u0947 \u0939\u0948\u0902,<\/strong><\/a> \u0907\u0938\u0915\u0947 \u0938\u093e\u0925 \u0914\u0930 \u092a\u093f\u091b\u0932\u0947 \u092a\u094d\u0930\u092e\u093e\u0923\u094b\u0902 \u0915\u0947 \u0938\u093e\u0925 \u0939\u092e \u0906\u0917\u0947 \u092c\u0939\u0941\u0924 \u0938\u093e\u0930\u0947 \u0938\u0940\u092e\u093e\u0913\u0902 \u0915\u0940 \u0917\u0923\u0928\u093e \u0932\u0917\u092d\u0917 \u0938\u0939\u091c \u0930\u0942\u092a \u0938\u0947 \u0915\u0930 \u092a\u093e\u090f\u0901\u0917\u0947\u0964<\/p>\n<p style=\"text-align: justify; \">\u092f\u0939 \u092a\u094d\u0930\u092e\u093e\u0923\u093f\u0924 \u0915\u0930\u0928\u093e \u0938\u0930\u0932 \u0939\u0948 \u0915\u093f <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to x_0}x = x_0<\/span><\/span>, \u0915\u094d\u092f\u094b\u0902\u0915\u093f \u0907\u0938\u0915\u0947 \u0938\u0924\u094d\u092f\u093e\u092a\u093f\u0924 \u0939\u094b\u0928\u0947 \u0915\u0947 \u0932\u093f\u090f \u092f\u0939 \u0906\u0935\u0936\u094d\u092f\u0915 \u0939\u0948 \u0915\u093f<\/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; \">\u0938\u0940\u092e\u093e \u0915\u0940 \u092a\u0930\u093f\u092d\u093e\u0937\u093e \u0915\u0947 \u0905\u0928\u0941\u0938\u093e\u0930, \u0939\u0930 \u090f\u092a\u094d\u092a\u094d\u0938\u093f\u0932\u0949\u0928 \u0915\u0947 \u0932\u093f\u090f \u0915\u092e \u0938\u0947 \u0915\u092e \u090f\u0915 \u0921\u0947\u0932\u094d\u091f\u093e \u0939\u094b\u0928\u093e \u091a\u093e\u0939\u093f\u090f \u091c\u094b \u0938\u092d\u0940 \u0905\u0928\u094d\u092f \u0936\u0930\u094d\u0924\u094b\u0902 \u0915\u094b \u092a\u0942\u0930\u093e \u0915\u0930\u0924\u093e \u0939\u0948; \u0907\u0938\u0932\u093f\u090f, \u0907\u0938\u0947 \u0938\u0924\u094d\u092f\u093e\u092a\u093f\u0924 \u0915\u0930\u0928\u0947 \u0915\u0947 \u0932\u093f\u090f \u0915\u0947\u0935\u0932 \u090f\u0915 \u0915\u094b \u0922\u0942\u0901\u0922\u0928\u093e \u092a\u0930\u094d\u092f\u093e\u092a\u094d\u0924 \u0939\u0948 \u0915\u093f \u0938\u0940\u092e\u093e \u0935\u0939\u0940 \u0939\u0948 \u091c\u094b \u0915\u0939\u093e \u091c\u093e \u0930\u0939\u093e \u0939\u0948\u0964 \u0932\u0947\u0915\u093f\u0928 \u092f\u0939 \u0935\u093e\u0938\u094d\u0924\u0935 \u092e\u0947\u0902 \u0938\u094d\u092a\u0937\u094d\u091f \u0939\u0948, \u0915\u094d\u092f\u094b\u0902\u0915\u093f \u0915\u094b\u0908 \u092d\u0940 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\delta\\leq\\epsilon<\/span><\/span> \u0907\u0938 \u0936\u0930\u094d\u0924 \u0915\u094b \u092a\u0942\u0930\u093e \u0915\u0930\u0947\u0917\u093e\u0964 \u0907\u0938\u0932\u093f\u090f: <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>\u0938\u0930\u0932 \u0938\u0940\u092e\u093e\u0913\u0902 \u0915\u0940 \u0917\u0923\u0928\u093e<\/h2>\n<p style=\"text-align: justify; \"><a href=\"https:\/\/www.youtube.com\/watch?v=FEPfoAfPsFY&amp;t=4155s\" rel=\"noopener\" target=\"_blank\"><strong>\u0939\u092e\u0928\u0947 \u091c\u093f\u0928 \u092a\u094d\u0930\u092e\u0947\u092f\u094b\u0902 \u0915\u0940 \u0938\u092e\u0940\u0915\u094d\u0937\u093e \u0915\u0940 \u0939\u0948<\/strong><\/a> \u0909\u0928\u0915\u0940 \u092e\u0926\u0926 \u0938\u0947 \u0939\u092e \u0938\u0940\u092e\u093e\u0913\u0902 \u0915\u0940 \u090f\u0915 \u0935\u093f\u0938\u094d\u0924\u0943\u0924 \u0936\u094d\u0930\u0943\u0902\u0916\u0932\u093e \u0915\u093e \u0938\u0939\u091c \u0930\u0942\u092a \u0938\u0947 \u0939\u093f\u0938\u093e\u092c \u0915\u0930 \u0938\u0915\u0924\u0947 \u0939\u0948\u0902, \u091c\u0948\u0938\u0947 \u0915\u093f \u0939\u092e \u092c\u0938 \u092b\u0932\u0928 \u0915\u093e \u092e\u093e\u0928 \u0928\u093f\u0915\u093e\u0932 \u0930\u0939\u0947 \u0939\u094b\u0902\u0964 \u092f\u0939\u093e\u0901 \u0915\u0941\u091b \u0909\u0926\u093e\u0939\u0930\u0923 \u0926\u0947\u0916 \u0938\u0915\u0924\u0947 \u0939\u0948\u0902:<\/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>\u0935\u093e\u0938\u094d\u0924\u0935\u093f\u0915 \u091a\u0930 \u0935\u093e\u0932\u0947 \u0915\u093e\u0930\u094d\u092f\u094b\u0902 \u0915\u0940 \u0938\u0940\u092e\u093e \u0938\u093e\u0930\u093e\u0902\u0936: \u0907\u0938 \u0915\u0915\u094d\u0937\u093e \u092e\u0947\u0902 \u090f\u0915 \u0935\u093e\u0938\u094d\u0924\u0935\u093f\u0915 \u091a\u0930 \u0935\u093e\u0932\u0947 \u0915\u093e\u0930\u094d\u092f\u094b\u0902 \u0915\u0940 \u0938\u0940\u092e\u093e \u0915\u0940 \u0914\u092a\u091a\u093e\u0930\u093f\u0915 \u092a\u0930\u093f\u092d\u093e\u0937\u093e \u0915\u0940 \u0917\u0939\u0930\u093e\u0908 \u0938\u0947 \u0938\u092e\u0940\u0915\u094d\u0937\u093e \u0915\u0940 \u091c\u093e\u0924\u0940 \u0939\u0948, \u0914\u0930 \u0907\u0938\u0915\u0947 \u0906\u0927\u093e\u0930 \u092a\u0930 \u0938\u0940\u092e\u093e\u0913\u0902 \u0915\u0947 \u092c\u0940\u091c\u0917\u0923\u093f\u0924 \u0915\u0940 \u092a\u094d\u0930\u092e\u0941\u0916 \u0917\u0941\u0923\u094b\u0902 \u0915\u094b \u092a\u094d\u0930\u0926\u0930\u094d\u0936\u093f\u0924 \u0915\u093f\u092f\u093e \u091c\u093e\u0924\u093e \u0939\u0948\u0964 \u0936\u093f\u0915\u094d\u0937\u0923 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