{"id":29345,"date":"2024-10-22T13:00:19","date_gmt":"2024-10-22T13:00:19","guid":{"rendered":"http:\/\/toposuranos.com\/material\/?p=29345"},"modified":"2024-10-23T14:37:26","modified_gmt":"2024-10-23T14:37:26","slug":"%e0%a4%85%e0%a4%a8%e0%a4%82%e0%a4%a4-%e0%a4%b8%e0%a5%80%e0%a4%ae%e0%a4%be-%e0%a4%aa%e0%a4%b0%e0%a4%bf%e0%a4%ad%e0%a4%be%e0%a4%b7%e0%a4%be%e0%a4%8f%e0%a4%81-%e0%a4%94%e0%a4%b0-%e0%a4%89%e0%a4%a6","status":"publish","type":"post","link":"https:\/\/toposuranos.com\/material\/hi\/%e0%a4%85%e0%a4%a8%e0%a4%82%e0%a4%a4-%e0%a4%b8%e0%a5%80%e0%a4%ae%e0%a4%be-%e0%a4%aa%e0%a4%b0%e0%a4%bf%e0%a4%ad%e0%a4%be%e0%a4%b7%e0%a4%be%e0%a4%8f%e0%a4%81-%e0%a4%94%e0%a4%b0-%e0%a4%89%e0%a4%a6\/","title":{"rendered":"\u0905\u0928\u0902\u0924 \u0938\u0940\u092e\u093e: \u092a\u0930\u093f\u092d\u093e\u0937\u093e\u090f\u0901 \u0914\u0930 \u0909\u0926\u093e\u0939\u0930\u0923"},"content":{"rendered":"<p><center><\/p>\n<h1>\u0905\u0928\u0902\u0924 \u0938\u0940\u092e\u093e: \u092a\u0930\u093f\u092d\u093e\u0937\u093e\u090f\u0901 \u0914\u0930 \u0909\u0926\u093e\u0939\u0930\u0923<\/h1>\n<p><em><\/p>\n<p><strong>\u0938\u093e\u0930\u093e\u0902\u0936:<\/strong><br \/>\n\u0907\u0938 \u0915\u0915\u094d\u0937\u093e \u092e\u0947\u0902 \u0905\u0928\u0902\u0924 \u0938\u0940\u092e\u093e\u0913\u0902 \u092a\u0930 \u091a\u0930\u094d\u091a\u093e \u0915\u0940 \u091c\u093e\u090f\u0917\u0940, \u091c\u094b <span class=\"katex-eq\" data-katex-display=\"false\">x<\/span> \u0915\u0947 \u0905\u0928\u0902\u0924 \u0915\u0940 \u0913\u0930 \u092c\u0922\u093c\u0928\u0947 \u092a\u0930 <span class=\"katex-eq\" data-katex-display=\"false\">f(x)<\/span> \u0915\u0947 \u0935\u094d\u092f\u0935\u0939\u093e\u0930 \u0915\u093e \u0935\u0930\u094d\u0923\u0928 \u0915\u0930\u0924\u0940 \u0939\u0948\u0964 <span class=\"katex-eq\" data-katex-display=\"false\">\\lim_{x\\to \\infty} \\frac{1}{x} = 0<\/span> \u0914\u0930 <span class=\"katex-eq\" data-katex-display=\"false\">\\lim_{x\\to \\infty} k = k<\/span> \u091c\u0948\u0938\u0947 \u092e\u0942\u0932\u092d\u0942\u0924 \u0938\u0940\u092e\u093e\u0913\u0902 \u0915\u094b \u0938\u092e\u091d\u093e\u092f\u093e \u091c\u093e\u090f\u0917\u093e, \u0938\u093e\u0925 \u0939\u0940 \u0938\u0940\u092e\u093f\u0924 \u0938\u0940\u092e\u093e\u0913\u0902 \u0915\u0947 \u0938\u092e\u093e\u0928 \u092c\u0940\u091c\u0917\u0923\u093f\u0924\u0940\u092f \u0917\u0941\u0923 \u092d\u0940 \u092a\u0947\u0936 \u0915\u093f\u090f \u091c\u093e\u090f\u0902\u0917\u0947\u0964\n<\/p>\n<p><\/em><\/p>\n<p><strong>\u0938\u0940\u0916\u0928\u0947 \u0915\u0947 \u0909\u0926\u094d\u0926\u0947\u0936\u094d\u092f:<\/strong><br \/>\n\u0915\u0915\u094d\u0937\u093e \u0938\u092e\u093e\u092a\u094d\u0924 \u0939\u094b\u0928\u0947 \u092a\u0930, \u091b\u093e\u0924\u094d\u0930 \u0938\u0915\u094d\u0937\u092e \u0939\u094b\u0902\u0917\u0947:<\/p>\n<ol style=\"text-align:left;\">\n<li><strong>\u0935\u0930\u094d\u0923\u0928<\/strong> \u0915\u0930\u0928\u093e \u0915\u093f \u091c\u092c <span class=\"katex-eq\" data-katex-display=\"false\">x<\/span> \u0905\u0928\u0902\u0924 \u0915\u0940 \u0913\u0930 \u092c\u0922\u093c\u0924\u093e \u0939\u0948 \u0924\u094b <span class=\"katex-eq\" data-katex-display=\"false\">f(x)<\/span> \u0915\u093e \u0935\u094d\u092f\u0935\u0939\u093e\u0930 \u0915\u094d\u092f\u093e \u0939\u094b\u0924\u093e \u0939\u0948\u0964<\/li>\n<li><strong>\u092a\u0930\u093f\u092d\u093e\u0937\u093f\u0924<\/strong> \u0915\u0930\u0928\u093e \u0915\u093f \u0917\u0923\u093f\u0924\u0940\u092f \u0938\u0902\u0915\u0947\u0924\u094b\u0902 \u0915\u093e \u0909\u092a\u092f\u094b\u0917 \u0915\u0930\u0915\u0947 \u0905\u0928\u0902\u0924 \u0938\u0940\u092e\u093e \u0915\u0948\u0938\u0947 \u0935\u094d\u092f\u0915\u094d\u0924 \u0915\u0940 \u091c\u093e\u0924\u0940 \u0939\u0948\u0964<\/li>\n<li><strong>\u0932\u093e\u0917\u0942<\/strong> \u0915\u0930\u0928\u093e \u0915\u093f \u0905\u0928\u0902\u0924 \u0938\u0940\u092e\u093e\u0913\u0902 \u0915\u0940 \u0917\u0923\u0928\u093e \u092e\u0947\u0902 \u092c\u0940\u091c\u0917\u0923\u093f\u0924\u0940\u092f \u0917\u0941\u0923\u094b\u0902 \u0915\u093e \u0915\u0948\u0938\u0947 \u0909\u092a\u092f\u094b\u0917 \u0915\u093f\u092f\u093e \u091c\u093e\u0924\u093e \u0939\u0948\u0964<\/li>\n<li><strong>\u0905\u0902\u0924\u0930<\/strong> \u0915\u0930\u0928\u093e \u0915\u093f \u0935\u093f\u092d\u093f\u0928\u094d\u0928 \u092e\u093e\u092e\u0932\u094b\u0902 \u092e\u0947\u0902 \u0905\u0928\u0902\u0924 \u0938\u0940\u092e\u093e\u0913\u0902 \u0915\u093e \u0935\u094d\u092f\u093e\u0935\u0939\u093e\u0930\u093f\u0915 \u0930\u0942\u092a \u0938\u0947 \u0909\u092a\u092f\u094b\u0917 \u0915\u0948\u0938\u0947 \u0915\u093f\u092f\u093e \u091c\u093e\u0924\u093e \u0939\u0948\u0964<\/li>\n<li><strong>\u0938\u093e\u092c\u093f\u0924<\/strong> \u0915\u0930\u0928\u093e \u0915\u093f \u0905\u0928\u0902\u0924 \u0938\u0940\u092e\u093e\u0913\u0902 \u0915\u0947 \u091c\u094b\u0921\u093c, \u0918\u091f\u093e\u0935, \u0917\u0941\u0923\u093e, \u092d\u093e\u0917, \u0914\u0930 \u0918\u093e\u0924 \u0915\u0947 \u0917\u0941\u0923 \u0935\u0948\u0927 \u0939\u0948\u0902\u0964<\/li>\n<li><strong>\u0939\u0932<\/strong> \u0915\u0930\u0928\u093e \u0915\u093f \u0915\u0948\u0938\u0947 \u0905\u0928\u0902\u0924 \u0938\u0940\u092e\u093e\u0913\u0902 \u0915\u0947 \u0935\u094d\u092f\u093e\u0935\u0939\u093e\u0930\u093f\u0915 \u092a\u094d\u0930\u0936\u094d\u0928\u094b\u0902 \u0915\u094b \u0935\u093f\u092d\u093f\u0928\u094d\u0928 \u0915\u093e\u0930\u094d\u092f\u094b\u0902 \u092e\u0947\u0902 \u0939\u0932 \u0915\u093f\u092f\u093e \u091c\u093e\u090f\u0964<\/li>\n<\/ol>\n<p><u>\u0935\u093f\u0937\u092f \u0938\u0942\u091a\u0940<\/u>:<br \/>\n<a href=\"#1\">\u092a\u0930\u093f\u091a\u092f<\/a><br \/>\n<a href=\"#2\">\u0905\u0928\u0902\u0924 \u0938\u0940\u092e\u093e \u0915\u0940 \u092a\u0930\u093f\u092d\u093e\u0937\u093e<\/a><br \/>\n<a href=\"#3\">\u0905\u0928\u0902\u0924 \u0938\u0940\u092e\u093e\u0913\u0902 \u0915\u0947 \u092e\u0942\u0932\u092d\u0942\u0924 \u0917\u0941\u0923<\/a><br \/>\n<a href=\"#4\">\u0905\u0928\u0902\u0924 \u0938\u0940\u092e\u093e\u0913\u0902 \u0915\u093e \u092c\u0940\u091c\u0917\u0923\u093f\u0924<\/a><br \/>\n<a href=\"#5\">\u0924\u0930\u094d\u0915\u0936\u0940\u0932 \u0915\u093e\u0930\u094d\u092f\u094b\u0902 \u092e\u0947\u0902 \u0905\u0928\u0902\u0924 \u0938\u0940\u092e\u093e<\/a><br \/>\n<a href=\"#6\">\u0905\u0928\u0902\u0924 \u0938\u0940\u092e\u093e\u0913\u0902 \u0915\u0947 \u0909\u0926\u093e\u0939\u0930\u0923<\/a><br \/>\n<\/center><\/p>\n<p><center><iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/MjjSAQLeNBE\" title=\"YouTube video player\" 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; color: #000000;\"><a href=\"https:\/\/www.youtube.com\/watch?v=MjjSAQLeNBE&amp;t=41s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">\u0915\u0932\u0928 \u0915\u0947 \u0938\u092c\u0938\u0947 \u0935\u093f\u0936\u093f\u0937\u094d\u091f \u0924\u0924\u094d\u0935\u094b\u0902 \u092e\u0947\u0902 \u0938\u0947 \u090f\u0915 \u0905\u0928\u0902\u0924 \u0914\u0930 \u0905\u0928\u0902\u0924 \u0938\u0940\u092e\u093e \u0939\u0948\u0964<\/span><\/strong><\/a> \u0905\u0928\u0902\u0924 \u0915\u093f\u0938\u0940 \u0935\u093f\u0936\u0947\u0937 \u0935\u093e\u0938\u094d\u0924\u0935\u093f\u0915 \u0938\u0902\u0916\u094d\u092f\u093e \u0915\u094b \u0928\u0939\u0940\u0902 \u0926\u0930\u094d\u0936\u093e\u0924\u093e \u0939\u0948, \u092c\u0932\u094d\u0915\u093f \u092f\u0939 \u090f\u0915 \u0910\u0938\u0940 \u092a\u0930\u093f\u092e\u093e\u0923 \u0915\u093e \u0935\u0930\u094d\u0923\u0928 \u0915\u0930\u0928\u0947 \u0915\u093e \u092a\u094d\u0930\u092f\u093e\u0938 \u0915\u0930\u0924\u093e \u0939\u0948 \u091c\u094b \u0915\u093f\u0938\u0940 \u092d\u0940 \u0935\u093e\u0938\u094d\u0924\u0935\u093f\u0915 \u0938\u0940\u092e\u093e \u0938\u0947 \u092a\u0930\u0947 \u0939\u0948\u0964 \u0909\u0926\u093e\u0939\u0930\u0923 \u0915\u0947 \u0932\u093f\u090f, \u091c\u092c \u0939\u092e\u093e\u0930\u0947 \u092a\u093e\u0938 <span class=\"katex-eq\" data-katex-display=\"false\">f(x) = 1\/x<\/span> \u092b\u093c\u0902\u0915\u094d\u0936\u0928 \u0939\u0948 \u0914\u0930 \u0939\u092e \u092a\u0942\u091b\u0924\u0947 \u0939\u0948\u0902 \u0915\u093f \u091c\u092c <span class=\"katex-eq\" data-katex-display=\"false\">x<\/span> \u091c\u093f\u0924\u0928\u093e \u0938\u0902\u092d\u0935 \u0939\u094b \u0909\u0924\u0928\u093e \u092c\u0921\u093c\u093e \u0939\u094b \u091c\u093e\u0924\u093e \u0939\u0948, \u0924\u092c <span class=\"katex-eq\" data-katex-display=\"false\">x<\/span> \u0905\u0928\u0902\u0924 \u0915\u0940 \u0913\u0930 \u092c\u0922\u093c\u0924\u093e \u0939\u0948 (<span class=\"katex-eq\" data-katex-display=\"false\">x\\to \\infty<\/span>), \u0924\u094b \u0939\u092e \u0926\u0947\u0916\u0924\u0947 \u0939\u0948\u0902 \u0915\u093f <span class=\"katex-eq\" data-katex-display=\"false\">f(x)<\/span> \u0936\u0942\u0928\u094d\u092f \u0915\u0940 \u0913\u0930 \u0905\u0928\u0902\u0924 \u0930\u0942\u092a \u0938\u0947 \u092a\u093e\u0938 \u0939\u094b \u0938\u0915\u0924\u093e \u0939\u0948\u0964 \u0907\u0938\u0915\u0947 \u0932\u093f\u090f \u0939\u092e \u0932\u093f\u0916\u0924\u0947 \u0939\u0948\u0902:<\/p>\n<p style=\"text-align: center; color: #000000;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to + \\infty}\\dfrac{1}{x} = 0<\/span>\n<p style=\"text-align: justify; color: #000000;\">\u0917\u094d\u0930\u093e\u092b\u093c\u093f\u0915 \u0930\u0942\u092a \u0938\u0947, \u092f\u0939 \u0907\u0938 \u092a\u094d\u0930\u0915\u093e\u0930 \u0926\u093f\u0916\u093e\u0908 \u0926\u0947\u0924\u093e \u0939\u0948:<\/p>\n<p><center><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/1.bp.blogspot.com\/-fX0lg2ICTTU\/YGD0hNJWI6I\/AAAAAAAAEwQ\/0v8hW6ARkQYDIzT_eG5WVgZ0-pPwwPBwgCLcBGAsYHQ\/s0\/limiteAlinfinito.PNG\" alt=\"limite al infinito\" class=\"alignnone size-full lazyload\" width=\"400\" height=\"300\" \/><noscript><img decoding=\"async\" src=\"https:\/\/1.bp.blogspot.com\/-fX0lg2ICTTU\/YGD0hNJWI6I\/AAAAAAAAEwQ\/0v8hW6ARkQYDIzT_eG5WVgZ0-pPwwPBwgCLcBGAsYHQ\/s0\/limiteAlinfinito.PNG\" alt=\"limite al infinito\" class=\"alignnone size-full lazyload\" width=\"400\" height=\"300\" \/><\/noscript><\/center><\/p>\n<p><a name=\"2\"><\/a><\/p>\n<h2>\u0905\u0928\u0902\u0924 \u0938\u0940\u092e\u093e \u0915\u0940 \u092a\u0930\u093f\u092d\u093e\u0937\u093e<\/h2>\n<p style=\"text-align: justify; color: #000000;\"><a href=\"https:\/\/www.youtube.com\/watch?v=MjjSAQLeNBE&amp;t=144s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">\u0905\u092d\u0940-\u0905\u092d\u0940 \u092a\u094d\u0930\u0938\u094d\u0924\u0941\u0924 \u0915\u093f\u090f \u0917\u090f \u0907\u0938 \u0935\u093f\u091a\u093e\u0930 \u0915\u0947 \u0906\u0927\u093e\u0930 \u092a\u0930<\/span><\/strong><\/a>, \u0939\u092e \u0905\u0928\u0902\u0924 \u0938\u0940\u092e\u093e \u0915\u0940 \u0917\u0923\u093f\u0924\u0940\u092f \u092a\u0930\u093f\u092d\u093e\u0937\u093e \u0924\u0948\u092f\u093e\u0930 \u0915\u0930 \u0938\u0915\u0924\u0947 \u0939\u0948\u0902:<\/p>\n<p style=\"text-align: justify; color: #000000;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to +\\infty}f(x) = L := (\\forall\\epsilon\\gt 0) (\\exists M\\in\\mathbb{R})(M\\lt x \\rightarrow |f(x) - L|\\lt \\epsilon )<\/span>\n<p style=\"text-align: justify; color: #000000;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to -\\infty}f(x) = L := (\\forall\\epsilon\\gt 0) (\\exists N\\in\\mathbb{R})(x\\lt N \\rightarrow |f(x) - L|\\lt \\epsilon )<\/span>\n<p style=\"text-align: justify; color: #000000;\">\u0907\u0938 \u0938\u0940\u092e\u093e \u0915\u0940 \u0938\u0939\u091c \u0927\u093e\u0930\u0923\u093e \u0939\u092e\u0947\u0902 \u092c\u0924\u093e\u0924\u0940 \u0939\u0948 \u0915\u093f \u091c\u092c <span class=\"katex-eq\" data-katex-display=\"false\">x<\/span> \u092e\u0942\u0932 \u0938\u0947 \u091c\u093f\u0924\u0928\u093e \u091a\u093e\u0939\u0947\u0902 \u0909\u0924\u0928\u093e \u0926\u0942\u0930 \u0939\u094b \u091c\u093e\u0924\u093e \u0939\u0948, \u091a\u093e\u0939\u0947 \u0935\u0939 \u0926\u093e\u090f\u0901 \u092f\u093e \u092c\u093e\u090f\u0901 \u0939\u094b, \u0924\u092c <span class=\"katex-eq\" data-katex-display=\"false\">f(x)<\/span> \u0915\u093e \u0915\u094d\u092f\u093e \u0939\u094b\u0924\u093e \u0939\u0948\u0964 \u0905\u0928\u0902\u0924 \u0938\u0940\u092e\u093e\u0913\u0902 \u0915\u0940 \u0917\u0923\u0928\u093e \u0915\u0947 \u0932\u093f\u090f \u0930\u0923\u0928\u0940\u0924\u093f \u092c\u0939\u0941\u0924 \u0905\u0927\u093f\u0915 \u092d\u093f\u0928\u094d\u0928 \u0928\u0939\u0940\u0902 \u0939\u0948 \u091c\u093f\u0924\u0928\u0940 \u0915\u093f \u0939\u092e \u0938\u0940\u092e\u093f\u0924 \u0938\u0940\u092e\u093e\u0913\u0902 \u0915\u0940 \u0917\u0923\u0928\u093e \u0915\u0947 \u0932\u093f\u090f \u0909\u092a\u092f\u094b\u0917 \u0915\u0930\u0924\u0947 \u0939\u0948\u0902, \u0915\u094d\u092f\u094b\u0902\u0915\u093f \u0907\u0938\u0915\u093e \u092c\u0940\u091c\u0917\u0923\u093f\u0924 \u0935\u094d\u092f\u093e\u0935\u0939\u093e\u0930\u093f\u0915 \u0930\u0942\u092a \u0938\u0947 \u0938\u092e\u093e\u0928 \u0939\u0948, \u0939\u092e\u0947\u0902 \u0915\u0947\u0935\u0932 \u0928\u093f\u092e\u094d\u0928\u0932\u093f\u0916\u093f\u0924 \u092a\u0930\u093f\u0923\u093e\u092e\u094b\u0902 \u092a\u0930 \u0927\u094d\u092f\u093e\u0928 \u0926\u0947\u0928\u093e \u0939\u094b\u0917\u093e:<\/p>\n<p><a name=\"3\"><\/a><\/p>\n<h2>\u0905\u0928\u0902\u0924 \u0938\u0940\u092e\u093e\u0913\u0902 \u0915\u0947 \u092e\u0942\u0932\u092d\u0942\u0924 \u0917\u0941\u0923<\/h2>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=MjjSAQLeNBE&amp;t=450s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">\u0907\u0928 \u092a\u0930\u093f\u092d\u093e\u0937\u093e\u0913\u0902 \u0915\u0947 \u0906\u0927\u093e\u0930 \u092a\u0930, \u0939\u092e \u0928\u093f\u092e\u094d\u0928\u0932\u093f\u0916\u093f\u0924 \u0938\u0940\u092e\u093e\u0913\u0902 \u0915\u094b \u092a\u094d\u0930\u092e\u093e\u0923\u093f\u0924 \u0915\u0930 \u0938\u0915\u0924\u0947 \u0939\u0948\u0902\u0964<\/span><\/strong><\/a><\/p>\n<ol style=\"text-align: justify; color: #000000;\">\n<li><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to \\pm\\infty}k = k <\/span><\/li>\n<li><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to \\pm\\infty}\\dfrac{1}{x} = 0 <\/span><\/li>\n<\/ol>\n<p style=\"text-align: justify; color: #000000;\"><span style=\"color: #000080;\">\u0938\u093f\u0926\u094d\u0927\u093e\u0902\u0924:<\/span><\/p>\n<ol style=\"text-align: justify; color: #000000;\">\n<li>\u0905\u0928\u0902\u0924 \u0938\u0940\u092e\u093e \u0915\u0940 \u092a\u0930\u093f\u092d\u093e\u0937\u093e \u0915\u0947 \u0905\u0928\u0941\u0938\u093e\u0930, \u0939\u092e\u0947\u0902 <span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to +\\infty}k = k <\/span> \u092a\u094d\u0930\u093e\u092a\u094d\u0924 \u0939\u094b\u0924\u093e \u0939\u0948, \u091c\u094b \u0915\u093f \u092f\u0939 \u0915\u0939\u0928\u0947 \u0915\u0947 \u0938\u092e\u093e\u0928 \u0939\u0948: <span class=\"katex-eq\" data-katex-display=\"false\">(\\forall\\epsilon\\gt 0) (\\exists M\\in\\mathbb{R})\\left(M\\lt x \\rightarrow \\left|k-k\\right|\\lt \\epsilon \\right)<\/span>\u0964 \u0932\u0947\u0915\u093f\u0928 <span class=\"katex-eq\" data-katex-display=\"false\">\\left|k-k\\right|=0\\lt \\epsilon <\/span> \u0915\u093f\u0938\u0940 \u092d\u0940 <span class=\"katex-eq\" data-katex-display=\"false\">\\epsilon \\gt 0<\/span> \u0915\u0947 \u0932\u093f\u090f \u0939\u092e\u0947\u0936\u093e \u0938\u0924\u094d\u092f \u0939\u0948, \u0914\u0930 <span class=\"katex-eq\" data-katex-display=\"false\">M<\/span> \u0915\u093e \u092e\u093e\u0928 \u0915\u0941\u091b \u092d\u0940 \u0939\u094b \u0938\u0915\u0924\u093e \u0939\u0948, \u0907\u0938\u0932\u093f\u090f \u0938\u0940\u092e\u093e \u0938\u0941\u0928\u093f\u0936\u094d\u091a\u093f\u0924 \u0939\u094b \u091c\u093e\u0924\u0940 \u0939\u0948\u0964\n<p>&nbsp;<\/li>\n<li>\u0939\u092e \u091c\u093e\u0928\u0924\u0947 \u0939\u0948\u0902 \u0915\u093f, \u092a\u0930\u093f\u092d\u093e\u0937\u093e \u0915\u0947 \u0905\u0928\u0941\u0938\u093e\u0930 <span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to +\\infty}k = k <\/span> \u0915\u093e \u092e\u0924\u0932\u092c \u0939\u0948: <span class=\"katex-eq\" data-katex-display=\"false\">(\\forall\\epsilon\\gt 0) (\\exists M\\in\\mathbb{R})\\left(M\\lt x \\rightarrow \\left|\\dfrac{1}{x}\\right|\\lt \\epsilon \\right)<\/span>\u0964 \u0932\u0947\u0915\u093f\u0928 \u092f\u0926\u093f \u0939\u092e <span class=\"katex-eq\" data-katex-display=\"false\">M=1\/\\epsilon<\/span> \u0932\u0947\u0924\u0947 \u0939\u0948\u0902, \u0924\u094b \u092f\u0939 \u0936\u0930\u094d\u0924 \u0924\u0941\u0930\u0902\u0924 \u0938\u0902\u0924\u0941\u0937\u094d\u091f \u0939\u094b \u091c\u093e\u0924\u0940 \u0939\u0948, \u091c\u093f\u0938\u0938\u0947 \u0938\u0940\u092e\u093e \u0938\u0941\u0928\u093f\u0936\u094d\u091a\u093f\u0924 \u0939\u094b \u091c\u093e\u0924\u0940 \u0939\u0948\u0964\n<p>&nbsp;<\/li>\n<\/ol>\n<p style=\"text-align: justify; color: #000000;\">\u0907\u0928 \u092a\u094d\u0930\u092e\u093e\u0923\u094b\u0902 \u0915\u094b <span class=\"katex-eq\" data-katex-display=\"false\">x\\to+\\infty<\/span> \u0915\u0947 \u0932\u093f\u090f \u092d\u0940 \u0938\u092e\u093e\u0928 \u0930\u0942\u092a \u0938\u0947 \u0915\u093f\u092f\u093e \u091c\u093e \u0938\u0915\u0924\u093e \u0939\u0948\u0964<\/p>\n<p><a name=\"4\"><\/a><\/p>\n<h2>\u0905\u0928\u0902\u0924 \u0938\u0940\u092e\u093e\u0913\u0902 \u0915\u093e \u092c\u0940\u091c\u0917\u0923\u093f\u0924<\/h2>\n<p style=\"text-align: justify; color: #000000;\"><a href=\"https:\/\/www.youtube.com\/watch?v=MjjSAQLeNBE&amp;t=620s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">\u0905\u0928\u0902\u0924 \u0938\u0940\u092e\u093e\u0913\u0902 \u0915\u093e \u092c\u0940\u091c\u0917\u0923\u093f\u0924 \u0938\u0940\u092e\u093f\u0924 \u0938\u0940\u092e\u093e\u0913\u0902 \u0915\u0947 \u092c\u0940\u091c\u0917\u0923\u093f\u0924 \u0915\u0947 \u0938\u092e\u093e\u0928 \u0939\u0948\u0964<\/span><\/strong><\/a> \u092f\u0926\u093f <span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to \\pm \\infty}f(x) = L<\/span> \u0914\u0930 <span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to \\pm \\infty}g(x) = M<\/span> \u0939\u0948, \u0924\u094b \u0928\u093f\u092e\u094d\u0928\u0932\u093f\u0916\u093f\u0924 \u0928\u093f\u092f\u092e \u0932\u093e\u0917\u0942 \u0939\u094b\u0924\u0947 \u0939\u0948\u0902:<\/p>\n<ol style=\"text-align: justify; color: #000000;\">\n<li><strong>\u0938\u0940\u092e\u093e\u0913\u0902 \u0915\u093e \u091c\u094b\u0921\u093c \u0914\u0930 \u0918\u091f\u093e\u0935:<\/strong> <span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to \\pm\\infty}(f(x)\\pm g(x)) = L \\pm M<\/span><\/li>\n<li><strong>\u0938\u093e\u0902\u0916\u094d\u092f\u093f\u0915 \u0917\u0941\u0923\u093e:<\/strong> <span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to \\pm\\infty}cf(x) = cL<\/span><\/li>\n<li><strong>\u0938\u0940\u092e\u093e\u0913\u0902 \u0915\u093e \u0917\u0941\u0923\u0928:<\/strong> <span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to \\pm\\infty}f(x)g(x) = LM<\/span><\/li>\n<li><strong>\u0938\u0940\u092e\u093e\u0913\u0902 \u0915\u093e \u092d\u093e\u0917:<\/strong> \u091c\u092c <span class=\"katex-eq\" data-katex-display=\"false\">M\\neq 0<\/span> \u0939\u094b, \u0924\u092c <span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to \\pm\\infty}f(x)\/g(x)=L\/M<\/span><\/li>\n<li><strong>\u0938\u0940\u092e\u093e\u0913\u0902 \u0915\u0940 \u0918\u093e\u0924:<\/strong> \u092f\u0926\u093f <span class=\"katex-eq\" data-katex-display=\"false\">p,q \\in\\mathbb{Z}<\/span> \u0914\u0930 <span class=\"katex-eq\" data-katex-display=\"false\">q\\neq 0<\/span> \u0939\u094b, \u0924\u094b <span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to \\pm\\infty}[f(x)]^{p\/q} = L^{p\/q}<\/span>\u0964 \u092f\u0926\u093f <span class=\"katex-eq\" data-katex-display=\"false\">q<\/span> \u0938\u092e \u0939\u0948, \u0924\u094b \u092f\u0939 \u092e\u093e\u0928 \u0932\u093f\u092f\u093e \u091c\u093e\u0924\u093e \u0939\u0948 \u0915\u093f <span class=\"katex-eq\" data-katex-display=\"false\">L\\geq 0<\/span> \u0939\u0948\u0964<\/li>\n<\/ol>\n<p style=\"text-align: justify; color: #000000;\">\u0935\u093e\u0938\u094d\u0924\u0935 \u092e\u0947\u0902, \u0907\u0928 \u0938\u092d\u0940 \u0917\u0941\u0923\u094b\u0902 \u0915\u0947 \u092a\u094d\u0930\u092e\u093e\u0923 <a href=\"https:\/\/toposuranos.com\/la-definicion-de-limite-demostraciones-y-teoremas\/\" target=\"_blank\" rel=\"noopener\">\u0938\u0940\u092e\u093f\u0924 \u0938\u0940\u092e\u093e\u0913\u0902<\/a> \u0915\u0947 \u092a\u094d\u0930\u092e\u093e\u0923 \u0915\u0947 \u0938\u092e\u093e\u0928 \u0939\u0948\u0902\u0964<\/p>\n<p><a name=\"5\"><\/a><\/p>\n<h2>\u0924\u0930\u094d\u0915\u0936\u0940\u0932 \u0915\u093e\u0930\u094d\u092f\u094b\u0902 \u092e\u0947\u0902 \u0905\u0928\u0902\u0924 \u0938\u0940\u092e\u093e<\/h2>\n<p style=\"text-align: justify; color: #000000;\"><a href=\"https:\/\/www.youtube.com\/watch?v=MjjSAQLeNBE&amp;t=792s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">\u090f\u0915 \u0924\u0930\u094d\u0915\u0936\u0940\u0932 \u0915\u093e\u0930\u094d\u092f \u0935\u0939 \u0939\u0948 \u091c\u093f\u0938\u0947 \u0926\u094b \u092c\u0939\u0941\u092a\u0926\u094b\u0902 \u0915\u0947 \u0905\u0928\u0941\u092a\u093e\u0924 \u0915\u0947 \u0930\u0942\u092a \u092e\u0947\u0902 \u0935\u094d\u092f\u0915\u094d\u0924 \u0915\u093f\u092f\u093e \u091c\u093e \u0938\u0915\u0924\u093e \u0939\u0948\u0964<\/span><\/strong><\/a> \u091c\u092c \u0907\u0938 \u092a\u094d\u0930\u0915\u093e\u0930 \u0915\u0947 \u0915\u093e\u0930\u094d\u092f \u092a\u0930 \u0905\u0928\u0902\u0924 \u0938\u0940\u092e\u093e\u0913\u0902 \u0915\u0940 \u0917\u0923\u0928\u093e \u0915\u0940 \u091c\u093e\u0924\u0940 \u0939\u0948, \u0924\u094b \u090f\u0915 \u0905\u0924\u094d\u092f\u0902\u0924 \u0909\u092a\u092f\u094b\u0917\u0940 \u0938\u0902\u092a\u0924\u094d\u0924\u093f \u0926\u0947\u0916\u0940 \u091c\u093e \u0938\u0915\u0924\u0940 \u0939\u0948:<\/p>\n<p style=\"text-align: justify; color: #000000;\">\u092e\u093e\u0928 \u0932\u0940\u091c\u093f\u090f \u0915\u093f \u0939\u092e\u0947\u0902 <span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to \\infty}P(x)\/Q(x)<\/span> \u0915\u0940 \u0917\u0923\u0928\u093e \u0915\u0930\u0928\u0940 \u0939\u0948:<\/p>\n<ul style=\"text-align: justify; color: #000000;\">\n<li>\u092f\u0926\u093f <span class=\"katex-eq\" data-katex-display=\"false\">P(x)<\/span> \u0915\u0940 \u0921\u093f\u0917\u094d\u0930\u0940 <span class=\"katex-eq\" data-katex-display=\"false\">Q(x)<\/span> \u0915\u0940 \u0921\u093f\u0917\u094d\u0930\u0940 \u0938\u0947 \u0905\u0927\u093f\u0915 \u0939\u0948, \u0924\u094b \u091c\u092c <span class=\"katex-eq\" data-katex-display=\"false\">x\\to\\infty<\/span>, \u0924\u092c <span class=\"katex-eq\" data-katex-display=\"false\">f(x)<\/span> \u0915\u0940 \u092a\u0930\u093f\u092e\u093e\u0923 \u0905\u0928\u0902\u0924 \u0939\u094b \u091c\u093e\u090f\u0917\u0940 (\u0938\u0940\u092e\u093e \u092e\u094c\u091c\u0942\u0926 \u0928\u0939\u0940\u0902 \u0939\u094b\u0917\u0940)\u0964<\/li>\n<li>\u091c\u092c <span class=\"katex-eq\" data-katex-display=\"false\">P(x)<\/span> \u0915\u0940 \u0921\u093f\u0917\u094d\u0930\u0940 <span class=\"katex-eq\" data-katex-display=\"false\">Q(x)<\/span> \u0915\u0940 \u0921\u093f\u0917\u094d\u0930\u0940 \u0938\u0947 \u0915\u092e \u0939\u094b, \u0924\u094b \u0938\u0940\u092e\u093e \u0936\u0942\u0928\u094d\u092f \u0939\u094b\u0917\u0940\u0964<\/li>\n<li>\u0914\u0930 \u0905\u0902\u0924 \u092e\u0947\u0902, \u092f\u0926\u093f <span class=\"katex-eq\" data-katex-display=\"false\">P(x)<\/span> \u0915\u0940 \u0921\u093f\u0917\u094d\u0930\u0940 <span class=\"katex-eq\" data-katex-display=\"false\">Q(x)<\/span> \u0915\u0940 \u0921\u093f\u0917\u094d\u0930\u0940 \u0915\u0947 \u092c\u0930\u093e\u092c\u0930 \u0939\u094b, \u0924\u094b \u0938\u0940\u092e\u093e \u0909\u0928 \u0936\u0940\u0930\u094d\u0937 \u0921\u093f\u0917\u094d\u0930\u0940 \u0917\u0941\u0923\u093e\u0902\u0915\u094b\u0902 \u0915\u0947 \u0905\u0928\u0941\u092a\u093e\u0924 \u0915\u0947 \u092c\u0930\u093e\u092c\u0930 \u0939\u094b\u0917\u0940\u0964<\/li>\n<\/ul>\n<p style=\"text-align: justify; color: #000000;\">\u0907\u0938 \u092a\u0930\u093f\u0923\u093e\u092e \u0915\u0940 \u0938\u092c\u0938\u0947 \u0905\u091a\u094d\u091b\u0940 \u092c\u093e\u0924 \u092f\u0939 \u0939\u0948 \u0915\u093f, \u091c\u0948\u0938\u093e \u0915\u093f \u0939\u092e \u0906\u0928\u0947 \u0935\u093e\u0932\u0947 \u0909\u0926\u093e\u0939\u0930\u0923\u094b\u0902 \u092e\u0947\u0902 \u0926\u0947\u0916\u0947\u0902\u0917\u0947, \u092f\u0939 \u0924\u092c \u092d\u0940 \u0915\u093e\u092e \u0915\u0930\u0924\u093e \u0939\u0948 \u091c\u092c \u0936\u093e\u092e\u093f\u0932 \u0918\u093e\u0924\u093e\u0902\u0915 \u092a\u0942\u0930\u094d\u0923\u093e\u0902\u0915 \u0928 \u0939\u094b\u0902\u0964<\/p>\n<p><a name=\"6\"><\/a><\/p>\n<h2>\u0905\u0928\u0902\u0924 \u0938\u0940\u092e\u093e\u0913\u0902 \u0915\u0947 \u0909\u0926\u093e\u0939\u0930\u0923<\/h2>\n<ol style=\"text-align: justify; color: #000000;\">\n<li><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to +\\infty}\\dfrac{x+1}{x^2+3}<\/span> <a href=\"https:\/\/www.youtube.com\/watch?v=MjjSAQLeNBE&amp;t=907s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">[\u0938\u092e\u093e\u0927\u093e\u0928]<\/span><\/strong><\/a><\/li>\n<li><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to -\\infty}\\dfrac{2x^3 + 7}{x^3 - x^2 + x + 7}<\/span> <a href=\"https:\/\/www.youtube.com\/watch?v=MjjSAQLeNBE&amp;t=986s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">[\u0938\u092e\u093e\u0927\u093e\u0928]<\/span><\/strong><\/a><\/li>\n<li><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to +\\infty}\\dfrac{9x^4 + x}{2x^4 + 5x^2 - x + 6}<\/span> <a href=\"https:\/\/www.youtube.com\/watch?v=MjjSAQLeNBE&amp;t=1049s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">[\u0938\u092e\u093e\u0927\u093e\u0928]<\/span><\/strong><\/a><\/li>\n<li><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to +\\infty}\\dfrac{10x^5 + x4 + 31}{x^4 - 7x^3 + 7x^2 + 9}<\/span> <a href=\"https:\/\/www.youtube.com\/watch?v=MjjSAQLeNBE&amp;t=1111s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">[\u0938\u092e\u093e\u0927\u093e\u0928]<\/span><\/strong><\/a><\/li>\n<li><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to +\\infty}\\dfrac{2\\sqrt{x}+x^{-1}}{3x - 7}<\/span> <a href=\"https:\/\/www.youtube.com\/watch?v=MjjSAQLeNBE&amp;t=1220s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">[\u0938\u092e\u093e\u0927\u093e\u0928]<\/span><\/strong><\/a><\/li>\n<li><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to -\\infty}\\dfrac{2x^{5\/3} - x^{1\/3} + 7}{x^{8\/5}+3x + \\sqrt{x}}<\/span> <a href=\"https:\/\/www.youtube.com\/watch?v=MjjSAQLeNBE&amp;t=1284s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">[\u0938\u092e\u093e\u0927\u093e\u0928]<\/span><\/strong><\/a><\/li>\n<li><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to +\\infty}\\dfrac{\\sqrt[3]{x}-5x+3}{2x + x^{2\/3} - 4}<\/span> <a href=\"https:\/\/www.youtube.com\/watch?v=MjjSAQLeNBE&amp;t=1406s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">[\u0938\u092e\u093e\u0927\u093e\u0928]<\/span><\/strong><\/a><\/li>\n<li><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to +\\infty}\\dfrac{x^{8\/3}+2x + \\sqrt{x}}{x^2+x-3}<\/span> <a href=\"https:\/\/www.youtube.com\/watch?v=MjjSAQLeNBE&amp;t=1521s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">[\u0938\u092e\u093e\u0927\u093e\u0928]<\/span><\/strong><\/a><\/li>\n<\/ol>\n","protected":false},"excerpt":{"rendered":"<p>\u0905\u0928\u0902\u0924 \u0938\u0940\u092e\u093e: \u092a\u0930\u093f\u092d\u093e\u0937\u093e\u090f\u0901 \u0914\u0930 \u0909\u0926\u093e\u0939\u0930\u0923 \u0938\u093e\u0930\u093e\u0902\u0936: \u0907\u0938 \u0915\u0915\u094d\u0937\u093e \u092e\u0947\u0902 \u0905\u0928\u0902\u0924 \u0938\u0940\u092e\u093e\u0913\u0902 \u092a\u0930 \u091a\u0930\u094d\u091a\u093e \u0915\u0940 \u091c\u093e\u090f\u0917\u0940, \u091c\u094b \u0915\u0947 \u0905\u0928\u0902\u0924 \u0915\u0940 \u0913\u0930 \u092c\u0922\u093c\u0928\u0947 \u092a\u0930 \u0915\u0947 \u0935\u094d\u092f\u0935\u0939\u093e\u0930 \u0915\u093e \u0935\u0930\u094d\u0923\u0928 \u0915\u0930\u0924\u0940 \u0939\u0948\u0964 \u0914\u0930 \u091c\u0948\u0938\u0947 \u092e\u0942\u0932\u092d\u0942\u0924 \u0938\u0940\u092e\u093e\u0913\u0902 \u0915\u094b \u0938\u092e\u091d\u093e\u092f\u093e \u091c\u093e\u090f\u0917\u093e, \u0938\u093e\u0925 \u0939\u0940 \u0938\u0940\u092e\u093f\u0924 \u0938\u0940\u092e\u093e\u0913\u0902 \u0915\u0947 \u0938\u092e\u093e\u0928 \u092c\u0940\u091c\u0917\u0923\u093f\u0924\u0940\u092f \u0917\u0941\u0923 \u092d\u0940 \u092a\u0947\u0936 \u0915\u093f\u090f \u091c\u093e\u090f\u0902\u0917\u0947\u0964 \u0938\u0940\u0916\u0928\u0947 \u0915\u0947 \u0909\u0926\u094d\u0926\u0947\u0936\u094d\u092f: \u0915\u0915\u094d\u0937\u093e \u0938\u092e\u093e\u092a\u094d\u0924 \u0939\u094b\u0928\u0947 \u092a\u0930, \u091b\u093e\u0924\u094d\u0930 [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":29336,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"iawp_total_views":5,"footnotes":""},"categories":[862,577],"tags":[],"class_list":["post-29345","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-862","category-577"],"yoast_head":"<!-- This site is optimized 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