{"id":29342,"date":"2024-10-22T13:00:33","date_gmt":"2024-10-22T13:00:33","guid":{"rendered":"http:\/\/toposuranos.com\/material\/?p=29342"},"modified":"2024-10-23T14:36:44","modified_gmt":"2024-10-23T14:36:44","slug":"%e6%97%a0%e7%a9%b7%e6%9e%81%e9%99%90%ef%bc%9a%e5%ae%9a%e4%b9%89%e4%b8%8e%e7%a4%ba%e4%be%8b","status":"publish","type":"post","link":"https:\/\/toposuranos.com\/material\/zh\/%e6%97%a0%e7%a9%b7%e6%9e%81%e9%99%90%ef%bc%9a%e5%ae%9a%e4%b9%89%e4%b8%8e%e7%a4%ba%e4%be%8b\/","title":{"rendered":"\u65e0\u7a77\u6781\u9650\uff1a\u5b9a\u4e49\u4e0e\u793a\u4f8b"},"content":{"rendered":"<p><center><\/p>\n<h1>\u65e0\u7a77\u6781\u9650\uff1a\u5b9a\u4e49\u4e0e\u793a\u4f8b<\/h1>\n<p><em><\/p>\n<p><strong>\u6458\u8981\uff1a<\/strong><br \/>\n\u672c\u8282\u8bfe\u5c06\u63a2\u8ba8\u65e0\u7a77\u6781\u9650\uff0c\u63cf\u8ff0\u5f53 <span class=\"katex-eq\" data-katex-display=\"false\">x<\/span> \u8d8b\u8fd1\u4e8e\u65e0\u7a77\u5927\u65f6 <span class=\"katex-eq\" data-katex-display=\"false\">f(x)<\/span> \u7684\u884c\u4e3a\u3002\u57fa\u7840\u6781\u9650\u5982 <span class=\"katex-eq\" data-katex-display=\"false\">\\lim_{x\\to \\infty} \\frac{1}{x} = 0<\/span> \u548c <span class=\"katex-eq\" data-katex-display=\"false\">\\lim_{x\\to \\infty} k = k<\/span> \u5c06\u88ab\u89e3\u91ca\uff0c\u540c\u65f6\u8fd8\u4f1a\u4ecb\u7ecd\u7c7b\u4f3c\u4e8e\u6709\u9650\u6781\u9650\u7684\u4ee3\u6570\u6027\u8d28\u3002\n<\/p>\n<p><\/em><\/p>\n<p><strong>\u5b66\u4e60\u76ee\u6807\uff1a<\/strong><br \/>\n\u5728\u672c\u8282\u8bfe\u7ed3\u675f\u65f6\uff0c\u5b66\u751f\u5c06\u80fd\u591f<\/p>\n<ol style=\"text-align:left;\">\n<li><strong>\u63cf\u8ff0<\/strong> \u5f53 <span class=\"katex-eq\" data-katex-display=\"false\">x<\/span> \u8d8b\u8fd1\u4e8e\u65e0\u7a77\u5927\u65f6 <span class=\"katex-eq\" data-katex-display=\"false\">f(x)<\/span> \u7684\u884c\u4e3a\u3002<\/li>\n<li><strong>\u5b9a\u4e49<\/strong> \u4f7f\u7528\u6570\u5b66\u7b26\u53f7\u8868\u793a\u7684\u65e0\u7a77\u6781\u9650\u3002<\/li>\n<li><strong>\u5e94\u7528<\/strong> \u4ee3\u6570\u6027\u8d28\u6765\u8ba1\u7b97\u65e0\u7a77\u6781\u9650\u3002<\/li>\n<li><strong>\u533a\u5206<\/strong> \u4e0d\u540c\u60c5\u51b5\u4e0b\u65e0\u7a77\u6781\u9650\u5728\u6709\u7406\u51fd\u6570\u4e2d\u7684\u8868\u73b0\u3002<\/li>\n<li><strong>\u8bc1\u660e<\/strong> \u65e0\u7a77\u6781\u9650\u7684\u52a0\u3001\u51cf\u3001\u4e58\u3001\u9664\u548c\u5e42\u7684\u6027\u8d28\u7684\u6709\u6548\u6027\u3002<\/li>\n<li><strong>\u89e3\u51b3<\/strong> \u65e0\u7a77\u6781\u9650\u5728\u4e0d\u540c\u51fd\u6570\u4e2d\u7684\u5b9e\u9645\u95ee\u9898\u3002<\/li>\n<\/ol>\n<p><u>\u5185\u5bb9\u76ee\u5f55<\/u>:<br \/>\n<a href=\"#1\">\u4ecb\u7ecd<\/a><br \/>\n<a href=\"#2\">\u65e0\u7a77\u6781\u9650\u7684\u5b9a\u4e49<\/a><br \/>\n<a href=\"#3\">\u65e0\u7a77\u6781\u9650\u7684\u57fa\u7840\u77e5\u8bc6<\/a><br \/>\n<a href=\"#4\">\u65e0\u7a77\u6781\u9650\u7684\u4ee3\u6570<\/a><br \/>\n<a href=\"#5\">\u6709\u7406\u51fd\u6570\u4e2d\u7684\u65e0\u7a77\u6781\u9650<\/a><br \/>\n<a href=\"#6\">\u65e0\u7a77\u6781\u9650\u7684\u793a\u4f8b<\/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>\u4ecb\u7ecd<\/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;\">\u5fae\u79ef\u5206\u4e2d\u6700\u5177\u4ee3\u8868\u6027\u7684\u5143\u7d20\u4e4b\u4e00\u662f\u65e0\u7a77\u548c\u65e0\u7a77\u6781\u9650\u3002<\/span><\/strong><\/a> \u65e0\u7a77\u5e76\u4e0d\u6307\u5177\u4f53\u7684\u5b9e\u6570\uff0c\u800c\u662f\u8bd5\u56fe\u63cf\u8ff0\u8d85\u51fa\u4efb\u4f55\u5b9e\u6570\u8303\u56f4\u7684\u91cf\u7ea7\u3002\u4f8b\u5982\uff0c\u5f53\u6211\u4eec\u6709\u51fd\u6570 <span class=\"katex-eq\" data-katex-display=\"false\">f(x) = 1\/x<\/span> \u5e76\u63a2\u8ba8\u5f53 <span class=\"katex-eq\" data-katex-display=\"false\">x<\/span> \u8d8b\u8fd1\u4e8e\u65e0\u7a77\u5927 (<span class=\"katex-eq\" data-katex-display=\"false\">x\\to \\infty<\/span>) \u65f6\u7684\u884c\u4e3a\u65f6\uff0c\u6211\u4eec\u89c2\u5bdf\u5230 <span class=\"katex-eq\" data-katex-display=\"false\">f(x)<\/span> \u5c06\u4f1a\u65e0\u9650\u63a5\u8fd1\u4e8e\u96f6\u3002\u56e0\u6b64\uff0c\u6211\u4eec\u5199\u4f5c\uff1a<\/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;\">\u4ece\u56fe\u5f62\u4e0a\u770b\uff0c\u8fd9\u79cd\u60c5\u51b5\u8868\u73b0\u5982\u4e0b\uff1a<\/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>\u65e0\u7a77\u6781\u9650\u7684\u5b9a\u4e49<\/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;\">\u57fa\u4e8e\u521a\u624d\u4ecb\u7ecd\u7684\u6982\u5ff5<\/span><\/strong><\/a>\uff0c\u6211\u4eec\u53ef\u4ee5\u7528\u6570\u5b66\u516c\u5f0f\u6765\u5b9a\u4e49\u65e0\u7a77\u6781\u9650\uff1a<\/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;\">\u65e0\u7a77\u6781\u9650\u7684\u76f4\u89c2\u6982\u5ff5\u662f\uff0c\u5f53 <span class=\"katex-eq\" data-katex-display=\"false\">x<\/span> \u4ece\u539f\u70b9\u65e0\u7a77\u8fdc\u79bb\u65f6\uff0c\u4e0d\u8bba\u5411\u53f3\u8fd8\u662f\u5411\u5de6\uff0c\u51fd\u6570 <span class=\"katex-eq\" data-katex-display=\"false\">f(x)<\/span> \u7684\u884c\u4e3a\u3002\u8ba1\u7b97\u65e0\u7a77\u6781\u9650\u7684\u7b56\u7565\u4e0e\u8ba1\u7b97\u6709\u9650\u6781\u9650\u7684\u65b9\u6cd5\u7c7b\u4f3c\uff0c\u56e0\u4e3a\u5176\u4ee3\u6570\u57fa\u672c\u76f8\u540c\uff0c\u53ea\u9700\u8003\u8651\u4ee5\u4e0b\u7ed3\u679c\uff1a<\/p>\n<p><a name=\"3\"><\/a><\/p>\n<h2>\u65e0\u7a77\u6781\u9650\u7684\u57fa\u7840\u77e5\u8bc6<\/h2>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=MjjSAQLeNBE&amp;t=450s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">\u57fa\u4e8e\u8fd9\u4e9b\u5b9a\u4e49\uff0c\u6211\u4eec\u53ef\u4ee5\u8bc1\u660e<\/span><\/strong><\/a>\u4ee5\u4e0b\u57fa\u7840\u6781\u9650\u3002<\/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;\">\u8bc1\u660e\uff1a<\/span><\/p>\n<ol style=\"text-align: justify; color: #000000;\">\n<li>\u6839\u636e\u65e0\u7a77\u6781\u9650\u7684\u5b9a\u4e49\uff0c\u6211\u4eec\u6709 <span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to +\\infty}k = k <\/span>\uff0c\u8fd9\u7b49\u4ef7\u4e8e\u8bf4\uff1a<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>\u3002\u4f46\u662f <span class=\"katex-eq\" data-katex-display=\"false\">\\left|k-k\\right|=0\\lt \\epsilon <\/span> \u5bf9\u4efb\u4f55 <span class=\"katex-eq\" data-katex-display=\"false\">\\epsilon \\gt 0,<\/span> \u5747\u6210\u7acb\uff0c\u65e0\u8bba <span class=\"katex-eq\" data-katex-display=\"false\">M<\/span> \u7684\u503c\u4e3a\u4f55\uff0c\u56e0\u6b64\u6781\u9650\u5f97\u5230\u4fdd\u8bc1\u3002\n<p>&nbsp;<\/li>\n<li>\u4f17\u6240\u5468\u77e5\uff0c\u6839\u636e\u5b9a\u4e49 <span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to +\\infty}k = k <\/span> \u7b49\u4ef7\u4e8e\u8bf4\uff1a<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>\u3002\u4f46\u5982\u679c\u6211\u4eec\u8003\u8651 <span class=\"katex-eq\" data-katex-display=\"false\">M=1\/\\epsilon,<\/span> \u5219\u8be5\u9690\u542b\u6761\u4ef6\u7acb\u5373\u6ee1\u8db3\uff0c\u56e0\u6b64\u6781\u9650\u5f97\u5230\u4fdd\u8bc1\u3002\n<p>&nbsp;<\/li>\n<\/ol>\n<p style=\"text-align: justify; color: #000000;\">\u8fd9\u4e9b\u8bc1\u660e\u5728 <span class=\"katex-eq\" data-katex-display=\"false\">x\\to+\\infty<\/span> \u65f6\u4e5f\u4ee5\u7c7b\u4f3c\u65b9\u5f0f\u8fdb\u884c\u3002<\/p>\n<p><a name=\"4\"><\/a><\/p>\n<h2>\u65e0\u7a77\u6781\u9650\u7684\u4ee3\u6570<\/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;\">\u65e0\u7a77\u6781\u9650\u7684\u4ee3\u6570\u7c7b\u4f3c\u4e8e\u6709\u9650\u6781\u9650\u7684\u4ee3\u6570\u3002<\/span><\/strong><\/a> \u5982\u679c <span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to \\pm \\infty}f(x) = L<\/span> \u4e14 <span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to \\pm \\infty}g(x) = M<\/span>\uff0c\u5219\u4ee5\u4e0b\u89c4\u5219\u9002\u7528\uff1a<\/p>\n<ol style=\"text-align: justify; color: #000000;\">\n<li><strong>\u6781\u9650\u7684\u52a0\u6cd5\u548c\u51cf\u6cd5\uff1a<\/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>\u4e58\u4ee5\u5e38\u6570\uff1a<\/strong> <span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to \\pm\\infty}cf(x) = cL<\/span><\/li>\n<li><strong>\u6781\u9650\u7684\u4e58\u79ef\uff1a<\/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>\u6781\u9650\u7684\u5546\uff1a<\/strong> \u53ea\u8981 <span class=\"katex-eq\" data-katex-display=\"false\">M\\neq 0,<\/span> \u5219 <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>\u6781\u9650\u7684\u5e42\uff1a<\/strong> \u5982\u679c <span class=\"katex-eq\" data-katex-display=\"false\">p,q \\in\\mathbb{Z}<\/span> \u4e14 <span class=\"katex-eq\" data-katex-display=\"false\">q\\neq 0<\/span>\uff0c\u5219 <span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to \\pm\\infty}[f(x)]^{p\/q} = L^{p\/q}<\/span>\u3002\u5982\u679c <span class=\"katex-eq\" data-katex-display=\"false\">q<\/span> \u662f\u5076\u6570\uff0c\u9ed8\u8ba4 <span class=\"katex-eq\" data-katex-display=\"false\">L\\geq 0<\/span><\/li>\n<\/ol>\n<p style=\"text-align: justify; color: #000000;\">\u5b9e\u9645\u4e0a\uff0c\u8fd9\u4e9b\u6027\u8d28\u7684\u8bc1\u660e\u4e0e <a href=\"https:\/\/toposuranos.com\/la-definicion-de-limite-demostraciones-y-teoremas\/\" target=\"_blank\" rel=\"noopener\">\u6709\u9650\u6781\u9650<\/a> \u7684\u8bc1\u660e\u7c7b\u4f3c\u3002<\/p>\n<p><a name=\"5\"><\/a><\/p>\n<h2>\u6709\u7406\u51fd\u6570\u4e2d\u7684\u65e0\u7a77\u6781\u9650<\/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;\">\u6709\u7406\u51fd\u6570\u662f\u53ef\u4ee5\u8868\u793a\u4e3a\u4e24\u4e2a\u591a\u9879\u5f0f\u7684\u5546\u7684\u51fd\u6570\u3002<\/span><\/strong><\/a> \u5728\u5bf9\u8fd9\u7c7b\u51fd\u6570\u8fdb\u884c\u65e0\u7a77\u6781\u9650\u8ba1\u7b97\u65f6\uff0c\u53ef\u4ee5\u89c2\u5bdf\u5230\u4e00\u4e2a\u975e\u5e38\u6709\u7528\u7684\u6027\u8d28\uff1a<\/p>\n<p style=\"text-align: justify; color: #000000;\">\u5047\u8bbe\u6211\u4eec\u8981\u8ba1\u7b97 <span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to \\infty}P(x)\/Q(x)<\/span>\n<ul style=\"text-align: justify; color: #000000;\">\n<li>\u5982\u679c <span class=\"katex-eq\" data-katex-display=\"false\">P(x)<\/span> \u7684\u6b21\u6570\u5927\u4e8e <span class=\"katex-eq\" data-katex-display=\"false\">Q(x)<\/span> \u7684\u6b21\u6570\uff0c\u5219\u5f53 <span class=\"katex-eq\" data-katex-display=\"false\">x\\to\\infty<\/span> \u65f6\uff0c\u51fd\u6570 <span class=\"katex-eq\" data-katex-display=\"false\">f(x)<\/span> \u7684\u5927\u5c0f\u5c06\u65e0\u9650\u589e\u957f\uff08\u6781\u9650\u4e0d\u5b58\u5728\uff09\u3002<\/li>\n<li>\u5f53 <span class=\"katex-eq\" data-katex-display=\"false\">P(x)<\/span> \u7684\u6b21\u6570\u5c0f\u4e8e <span class=\"katex-eq\" data-katex-display=\"false\">Q(x)<\/span> \u7684\u6b21\u6570\u65f6\uff0c\u6781\u9650\u5c06\u4e3a\u96f6\u3002<\/li>\n<li>\u6700\u540e\uff0c\u5982\u679c <span class=\"katex-eq\" data-katex-display=\"false\">P(x)<\/span> \u7684\u6b21\u6570\u7b49\u4e8e <span class=\"katex-eq\" data-katex-display=\"false\">Q(x)<\/span> \u7684\u6b21\u6570\uff0c\u5219\u6781\u9650\u5c06\u7b49\u4e8e\u6700\u9ad8\u6b21\u6570\u9879\u7684\u7cfb\u6570\u7684\u5546\u3002<\/li>\n<\/ul>\n<p style=\"text-align: justify; color: #000000;\">\u8fd9\u4e00\u7ed3\u679c\u7684\u597d\u5904\u5728\u4e8e\uff0c\u5982\u6211\u4eec\u5c06\u5728\u63a5\u4e0b\u6765\u7684\u793a\u4f8b\u4e2d\u770b\u5230\u7684\uff0c\u5373\u4f7f\u6d89\u53ca\u7684\u5e42\u4e0d\u662f\u6574\u6570\uff0c\u4e5f\u80fd\u4ee5\u7c7b\u4f3c\u65b9\u5f0f\u8d77\u4f5c\u7528\u3002<\/p>\n<p><a name=\"6\"><\/a><\/p>\n<h2>\u65e0\u7a77\u6781\u9650\u7684\u793a\u4f8b<\/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;\">[\u89e3\u7b54]<\/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;\">[\u89e3\u7b54]<\/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;\">[\u89e3\u7b54]<\/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;\">[\u89e3\u7b54]<\/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;\">[\u89e3\u7b54]<\/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;\">[\u89e3\u7b54]<\/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;\">[\u89e3\u7b54]<\/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;\">[\u89e3\u7b54]<\/span><\/strong><\/a><\/li>\n<\/ol>\n","protected":false},"excerpt":{"rendered":"<p>\u65e0\u7a77\u6781\u9650\uff1a\u5b9a\u4e49\u4e0e\u793a\u4f8b \u6458\u8981\uff1a \u672c\u8282\u8bfe\u5c06\u63a2\u8ba8\u65e0\u7a77\u6781\u9650\uff0c\u63cf\u8ff0\u5f53 \u8d8b\u8fd1\u4e8e\u65e0\u7a77\u5927\u65f6 \u7684\u884c\u4e3a\u3002\u57fa\u7840\u6781\u9650\u5982 \u548c \u5c06\u88ab\u89e3\u91ca\uff0c\u540c\u65f6\u8fd8\u4f1a\u4ecb\u7ecd\u7c7b\u4f3c\u4e8e\u6709\u9650\u6781\u9650\u7684\u4ee3\u6570\u6027\u8d28\u3002 \u5b66\u4e60\u76ee\u6807\uff1a \u5728\u672c\u8282\u8bfe\u7ed3\u675f\u65f6\uff0c\u5b66\u751f\u5c06\u80fd\u591f \u63cf\u8ff0 \u5f53 \u8d8b\u8fd1\u4e8e\u65e0\u7a77\u5927\u65f6 \u7684\u884c\u4e3a\u3002 \u5b9a\u4e49 \u4f7f\u7528\u6570\u5b66\u7b26\u53f7\u8868\u793a\u7684\u65e0\u7a77\u6781\u9650\u3002 \u5e94\u7528 \u4ee3\u6570\u6027\u8d28\u6765\u8ba1\u7b97\u65e0\u7a77\u6781\u9650\u3002 \u533a\u5206 \u4e0d\u540c\u60c5\u51b5\u4e0b\u65e0\u7a77\u6781\u9650\u5728\u6709\u7406\u51fd\u6570\u4e2d\u7684\u8868\u73b0\u3002 \u8bc1\u660e \u65e0\u7a77\u6781\u9650\u7684\u52a0\u3001\u51cf\u3001\u4e58\u3001\u9664\u548c\u5e42\u7684\u6027\u8d28\u7684\u6709\u6548\u6027\u3002 \u89e3\u51b3 \u65e0\u7a77\u6781\u9650\u5728\u4e0d\u540c\u51fd\u6570\u4e2d\u7684\u5b9e\u9645\u95ee\u9898\u3002 \u5185\u5bb9\u76ee\u5f55: \u4ecb\u7ecd \u65e0\u7a77\u6781\u9650\u7684\u5b9a\u4e49 \u65e0\u7a77\u6781\u9650\u7684\u57fa\u7840\u77e5\u8bc6 \u65e0\u7a77\u6781\u9650\u7684\u4ee3\u6570 \u6709\u7406\u51fd\u6570\u4e2d\u7684\u65e0\u7a77\u6781\u9650 \u65e0\u7a77\u6781\u9650\u7684\u793a\u4f8b \u4ecb\u7ecd \u5fae\u79ef\u5206\u4e2d\u6700\u5177\u4ee3\u8868\u6027\u7684\u5143\u7d20\u4e4b\u4e00\u662f\u65e0\u7a77\u548c\u65e0\u7a77\u6781\u9650\u3002 \u65e0\u7a77\u5e76\u4e0d\u6307\u5177\u4f53\u7684\u5b9e\u6570\uff0c\u800c\u662f\u8bd5\u56fe\u63cf\u8ff0\u8d85\u51fa\u4efb\u4f55\u5b9e\u6570\u8303\u56f4\u7684\u91cf\u7ea7\u3002\u4f8b\u5982\uff0c\u5f53\u6211\u4eec\u6709\u51fd\u6570 \u5e76\u63a2\u8ba8\u5f53 \u8d8b\u8fd1\u4e8e\u65e0\u7a77\u5927 () \u65f6\u7684\u884c\u4e3a\u65f6\uff0c\u6211\u4eec\u89c2\u5bdf\u5230 \u5c06\u4f1a\u65e0\u9650\u63a5\u8fd1\u4e8e\u96f6\u3002\u56e0\u6b64\uff0c\u6211\u4eec\u5199\u4f5c\uff1a \u4ece\u56fe\u5f62\u4e0a\u770b\uff0c\u8fd9\u79cd\u60c5\u51b5\u8868\u73b0\u5982\u4e0b\uff1a \u65e0\u7a77\u6781\u9650\u7684\u5b9a\u4e49 \u57fa\u4e8e\u521a\u624d\u4ecb\u7ecd\u7684\u6982\u5ff5\uff0c\u6211\u4eec\u53ef\u4ee5\u7528\u6570\u5b66\u516c\u5f0f\u6765\u5b9a\u4e49\u65e0\u7a77\u6781\u9650\uff1a \u65e0\u7a77\u6781\u9650\u7684\u76f4\u89c2\u6982\u5ff5\u662f\uff0c\u5f53 \u4ece\u539f\u70b9\u65e0\u7a77\u8fdc\u79bb\u65f6\uff0c\u4e0d\u8bba\u5411\u53f3\u8fd8\u662f\u5411\u5de6\uff0c\u51fd\u6570 \u7684\u884c\u4e3a\u3002\u8ba1\u7b97\u65e0\u7a77\u6781\u9650\u7684\u7b56\u7565\u4e0e\u8ba1\u7b97\u6709\u9650\u6781\u9650\u7684\u65b9\u6cd5\u7c7b\u4f3c\uff0c\u56e0\u4e3a\u5176\u4ee3\u6570\u57fa\u672c\u76f8\u540c\uff0c\u53ea\u9700\u8003\u8651\u4ee5\u4e0b\u7ed3\u679c\uff1a \u65e0\u7a77\u6781\u9650\u7684\u57fa\u7840\u77e5\u8bc6 \u57fa\u4e8e\u8fd9\u4e9b\u5b9a\u4e49\uff0c\u6211\u4eec\u53ef\u4ee5\u8bc1\u660e\u4ee5\u4e0b\u57fa\u7840\u6781\u9650\u3002 \u8bc1\u660e\uff1a \u6839\u636e\u65e0\u7a77\u6781\u9650\u7684\u5b9a\u4e49\uff0c\u6211\u4eec\u6709 \uff0c\u8fd9\u7b49\u4ef7\u4e8e\u8bf4\uff1a\u3002\u4f46\u662f \u5bf9\u4efb\u4f55 \u5747\u6210\u7acb\uff0c\u65e0\u8bba \u7684\u503c\u4e3a\u4f55\uff0c\u56e0\u6b64\u6781\u9650\u5f97\u5230\u4fdd\u8bc1\u3002 &nbsp; \u4f17\u6240\u5468\u77e5\uff0c\u6839\u636e\u5b9a\u4e49 \u7b49\u4ef7\u4e8e\u8bf4\uff1a\u3002\u4f46\u5982\u679c\u6211\u4eec\u8003\u8651 [&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":72,"footnotes":""},"categories":[858,575],"tags":[],"class_list":["post-29342","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-858","category-575"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v28.0 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>\u65e0\u7a77\u6781\u9650\uff1a\u5b9a\u4e49\u4e0e\u793a\u4f8b - toposuranos.com\/material<\/title>\n<meta name=\"description\" content=\"\u4e0d\u61c2\u65e0\u7a77\u6781\u9650\u8ba9\u4f60\u5931\u5206\uff01\u5feb\u6765\u53d1\u73b0\u6700\u4ee4\u4eba\u60ca\u8bb6\u7684\u6280\u5de7\uff0c\u77ac\u95f4\u638c\u63e1\u5b83\u4eec\u3002\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/toposuranos.com\/material\/zh\/\u65e0\u7a77\u6781\u9650\uff1a\u5b9a\u4e49\u4e0e\u793a\u4f8b\/\" \/>\n<meta property=\"og:locale\" content=\"es_ES\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"\u65e0\u7a77\u6781\u9650\uff1a\u5b9a\u4e49\u4e0e\u793a\u4f8b\" \/>\n<meta property=\"og:description\" content=\"\u4e0d\u61c2\u65e0\u7a77\u6781\u9650\u8ba9\u4f60\u5931\u5206\uff01\u5feb\u6765\u53d1\u73b0\u6700\u4ee4\u4eba\u60ca\u8bb6\u7684\u6280\u5de7\uff0c\u77ac\u95f4\u638c\u63e1\u5b83\u4eec\u3002\" \/>\n<meta property=\"og:url\" content=\"https:\/\/toposuranos.com\/material\/zh\/\u65e0\u7a77\u6781\u9650\uff1a\u5b9a\u4e49\u4e0e\u793a\u4f8b\/\" \/>\n<meta property=\"og:site_name\" content=\"toposuranos.com\/material\" \/>\n<meta property=\"article:publisher\" content=\"https:\/\/www.facebook.com\/groups\/toposuranos\" \/>\n<meta property=\"article:published_time\" content=\"2024-10-22T13:00:33+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2024-10-23T14:36:44+00:00\" \/>\n<meta property=\"og:image\" content=\"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2024\/10\/limite-1024x585.jpg\" \/>\n<meta name=\"author\" content=\"giorgio.reveco\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:title\" content=\"\u65e0\u7a77\u6781\u9650\uff1a\u5b9a\u4e49\u4e0e\u793a\u4f8b\" \/>\n<meta name=\"twitter:description\" content=\"\u4e0d\u61c2\u65e0\u7a77\u6781\u9650\u8ba9\u4f60\u5931\u5206\uff01\u5feb\u6765\u53d1\u73b0\u6700\u4ee4\u4eba\u60ca\u8bb6\u7684\u6280\u5de7\uff0c\u77ac\u95f4\u638c\u63e1\u5b83\u4eec\u3002\" \/>\n<meta name=\"twitter:image\" content=\"https:\/\/toposuranos.com\/material\/wp-content\/uploads\/2024\/10\/limite.jpg\" \/>\n<meta name=\"twitter:creator\" content=\"@topuranos\" \/>\n<meta name=\"twitter:site\" content=\"@topuranos\" \/>\n<meta name=\"twitter:label1\" content=\"Escrito por\" \/>\n\t<meta name=\"twitter:data1\" content=\"giorgio.reveco\" \/>\n\t<meta name=\"twitter:label2\" content=\"Tiempo de lectura\" \/>\n\t<meta name=\"twitter:data2\" content=\"2 minutos\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\\\/\\\/schema.org\",\"@graph\":[{\"@type\":\"Article\",\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/zh\\\/%e6%97%a0%e7%a9%b7%e6%9e%81%e9%99%90%ef%bc%9a%e5%ae%9a%e4%b9%89%e4%b8%8e%e7%a4%ba%e4%be%8b\\\/#article\",\"isPartOf\":{\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/zh\\\/%e6%97%a0%e7%a9%b7%e6%9e%81%e9%99%90%ef%bc%9a%e5%ae%9a%e4%b9%89%e4%b8%8e%e7%a4%ba%e4%be%8b\\\/\"},\"author\":{\"name\":\"giorgio.reveco\",\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/#\\\/schema\\\/person\\\/e15164361c3f9a2a02cf6c234cf7fdc1\"},\"headline\":\"\u65e0\u7a77\u6781\u9650\uff1a\u5b9a\u4e49\u4e0e\u793a\u4f8b\",\"datePublished\":\"2024-10-22T13:00:33+00:00\",\"dateModified\":\"2024-10-23T14:36:44+00:00\",\"mainEntityOfPage\":{\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/zh\\\/%e6%97%a0%e7%a9%b7%e6%9e%81%e9%99%90%ef%bc%9a%e5%ae%9a%e4%b9%89%e4%b8%8e%e7%a4%ba%e4%be%8b\\\/\"},\"wordCount\":494,\"commentCount\":0,\"publisher\":{\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/#organization\"},\"image\":{\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/zh\\\/%e6%97%a0%e7%a9%b7%e6%9e%81%e9%99%90%ef%bc%9a%e5%ae%9a%e4%b9%89%e4%b8%8e%e7%a4%ba%e4%be%8b\\\/#primaryimage\"},\"thumbnailUrl\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/wp-content\\\/uploads\\\/2024\\\/10\\\/limite.jpg\",\"articleSection\":[\"\u5fae\u5206\",\"\u6570\u5b66\"],\"inLanguage\":\"es\",\"potentialAction\":[{\"@type\":\"CommentAction\",\"name\":\"Comment\",\"target\":[\"https:\\\/\\\/toposuranos.com\\\/material\\\/zh\\\/%e6%97%a0%e7%a9%b7%e6%9e%81%e9%99%90%ef%bc%9a%e5%ae%9a%e4%b9%89%e4%b8%8e%e7%a4%ba%e4%be%8b\\\/#respond\"]}]},{\"@type\":\"WebPage\",\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/zh\\\/%e6%97%a0%e7%a9%b7%e6%9e%81%e9%99%90%ef%bc%9a%e5%ae%9a%e4%b9%89%e4%b8%8e%e7%a4%ba%e4%be%8b\\\/\",\"url\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/zh\\\/%e6%97%a0%e7%a9%b7%e6%9e%81%e9%99%90%ef%bc%9a%e5%ae%9a%e4%b9%89%e4%b8%8e%e7%a4%ba%e4%be%8b\\\/\",\"name\":\"\u65e0\u7a77\u6781\u9650\uff1a\u5b9a\u4e49\u4e0e\u793a\u4f8b - toposuranos.com\\\/material\",\"isPartOf\":{\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/#website\"},\"primaryImageOfPage\":{\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/zh\\\/%e6%97%a0%e7%a9%b7%e6%9e%81%e9%99%90%ef%bc%9a%e5%ae%9a%e4%b9%89%e4%b8%8e%e7%a4%ba%e4%be%8b\\\/#primaryimage\"},\"image\":{\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/zh\\\/%e6%97%a0%e7%a9%b7%e6%9e%81%e9%99%90%ef%bc%9a%e5%ae%9a%e4%b9%89%e4%b8%8e%e7%a4%ba%e4%be%8b\\\/#primaryimage\"},\"thumbnailUrl\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/wp-content\\\/uploads\\\/2024\\\/10\\\/limite.jpg\",\"datePublished\":\"2024-10-22T13:00:33+00:00\",\"dateModified\":\"2024-10-23T14:36:44+00:00\",\"description\":\"\u4e0d\u61c2\u65e0\u7a77\u6781\u9650\u8ba9\u4f60\u5931\u5206\uff01\u5feb\u6765\u53d1\u73b0\u6700\u4ee4\u4eba\u60ca\u8bb6\u7684\u6280\u5de7\uff0c\u77ac\u95f4\u638c\u63e1\u5b83\u4eec\u3002\",\"breadcrumb\":{\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/zh\\\/%e6%97%a0%e7%a9%b7%e6%9e%81%e9%99%90%ef%bc%9a%e5%ae%9a%e4%b9%89%e4%b8%8e%e7%a4%ba%e4%be%8b\\\/#breadcrumb\"},\"inLanguage\":\"es\",\"potentialAction\":[{\"@type\":\"ReadAction\",\"target\":[\"https:\\\/\\\/toposuranos.com\\\/material\\\/zh\\\/%e6%97%a0%e7%a9%b7%e6%9e%81%e9%99%90%ef%bc%9a%e5%ae%9a%e4%b9%89%e4%b8%8e%e7%a4%ba%e4%be%8b\\\/\"]}]},{\"@type\":\"ImageObject\",\"inLanguage\":\"es\",\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/zh\\\/%e6%97%a0%e7%a9%b7%e6%9e%81%e9%99%90%ef%bc%9a%e5%ae%9a%e4%b9%89%e4%b8%8e%e7%a4%ba%e4%be%8b\\\/#primaryimage\",\"url\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/wp-content\\\/uploads\\\/2024\\\/10\\\/limite.jpg\",\"contentUrl\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/wp-content\\\/uploads\\\/2024\\\/10\\\/limite.jpg\",\"width\":1792,\"height\":1024},{\"@type\":\"BreadcrumbList\",\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/zh\\\/%e6%97%a0%e7%a9%b7%e6%9e%81%e9%99%90%ef%bc%9a%e5%ae%9a%e4%b9%89%e4%b8%8e%e7%a4%ba%e4%be%8b\\\/#breadcrumb\",\"itemListElement\":[{\"@type\":\"ListItem\",\"position\":1,\"name\":\"Portada\",\"item\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/es\\\/cursos-de-matematica-y-fisica\\\/\"},{\"@type\":\"ListItem\",\"position\":2,\"name\":\"\u65e0\u7a77\u6781\u9650\uff1a\u5b9a\u4e49\u4e0e\u793a\u4f8b\"}]},{\"@type\":\"WebSite\",\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/#website\",\"url\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/\",\"name\":\"toposuranos.com\\\/material\",\"description\":\"\",\"publisher\":{\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/#organization\"},\"potentialAction\":[{\"@type\":\"SearchAction\",\"target\":{\"@type\":\"EntryPoint\",\"urlTemplate\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/?s={search_term_string}\"},\"query-input\":{\"@type\":\"PropertyValueSpecification\",\"valueRequired\":true,\"valueName\":\"search_term_string\"}}],\"inLanguage\":\"es\"},{\"@type\":\"Organization\",\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/#organization\",\"name\":\"toposuranos.com\\\/material\",\"url\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/\",\"logo\":{\"@type\":\"ImageObject\",\"inLanguage\":\"es\",\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/#\\\/schema\\\/logo\\\/image\\\/\",\"url\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/wp-content\\\/uploads\\\/2023\\\/10\\\/logo.png\",\"contentUrl\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/wp-content\\\/uploads\\\/2023\\\/10\\\/logo.png\",\"width\":2400,\"height\":2059,\"caption\":\"toposuranos.com\\\/material\"},\"image\":{\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/#\\\/schema\\\/logo\\\/image\\\/\"},\"sameAs\":[\"https:\\\/\\\/www.facebook.com\\\/groups\\\/toposuranos\",\"https:\\\/\\\/x.com\\\/topuranos\",\"https:\\\/\\\/www.youtube.com\\\/channel\\\/UC16yDm12cPcrwsE0fAM7X1g\",\"https:\\\/\\\/www.linkedin.com\\\/company\\\/69429190\"]},{\"@type\":\"Person\",\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/#\\\/schema\\\/person\\\/e15164361c3f9a2a02cf6c234cf7fdc1\",\"name\":\"giorgio.reveco\",\"image\":{\"@type\":\"ImageObject\",\"inLanguage\":\"es\",\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/wp-content\\\/uploads\\\/2023\\\/10\\\/1694478625378-96x96.jpeg\",\"url\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/wp-content\\\/uploads\\\/2023\\\/10\\\/1694478625378-96x96.jpeg\",\"contentUrl\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/wp-content\\\/uploads\\\/2023\\\/10\\\/1694478625378-96x96.jpeg\",\"caption\":\"giorgio.reveco\"},\"description\":\"Soy Licenciado en F\u00edsica, Magister en Ingenier\u00eda Industrial y Docente Universitario. Me dedico a desmitificar la f\u00edsica y las matem\u00e1ticas. Mi objetivo es hacer que estos campos sean f\u00e1cilmente comprensibles para todos, proporcionando las herramientas para explorar no solo el mundo que nos rodea, sino tambi\u00e9n las profundidades de nuestra propia existencia y el orden natural que nos conecta con el cosmos.\",\"sameAs\":[\"http:\\\/\\\/toposuranos.com\\\/material\"],\"url\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/author\\\/giorgio-reveco\\\/\"}]}<\/script>\n<!-- \/ Yoast SEO plugin. -->","yoast_head_json":{"title":"\u65e0\u7a77\u6781\u9650\uff1a\u5b9a\u4e49\u4e0e\u793a\u4f8b - toposuranos.com\/material","description":"\u4e0d\u61c2\u65e0\u7a77\u6781\u9650\u8ba9\u4f60\u5931\u5206\uff01\u5feb\u6765\u53d1\u73b0\u6700\u4ee4\u4eba\u60ca\u8bb6\u7684\u6280\u5de7\uff0c\u77ac\u95f4\u638c\u63e1\u5b83\u4eec\u3002","robots":{"index":"index","follow":"follow","max-snippet":"max-snippet:-1","max-image-preview":"max-image-preview:large","max-video-preview":"max-video-preview:-1"},"canonical":"https:\/\/toposuranos.com\/material\/zh\/\u65e0\u7a77\u6781\u9650\uff1a\u5b9a\u4e49\u4e0e\u793a\u4f8b\/","og_locale":"es_ES","og_type":"article","og_title":"\u65e0\u7a77\u6781\u9650\uff1a\u5b9a\u4e49\u4e0e\u793a\u4f8b","og_description":"\u4e0d\u61c2\u65e0\u7a77\u6781\u9650\u8ba9\u4f60\u5931\u5206\uff01\u5feb\u6765\u53d1\u73b0\u6700\u4ee4\u4eba\u60ca\u8bb6\u7684\u6280\u5de7\uff0c\u77ac\u95f4\u638c\u63e1\u5b83\u4eec\u3002","og_url":"https:\/\/toposuranos.com\/material\/zh\/\u65e0\u7a77\u6781\u9650\uff1a\u5b9a\u4e49\u4e0e\u793a\u4f8b\/","og_site_name":"toposuranos.com\/material","article_publisher":"https:\/\/www.facebook.com\/groups\/toposuranos","article_published_time":"2024-10-22T13:00:33+00:00","article_modified_time":"2024-10-23T14:36:44+00:00","og_image":[{"url":"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2024\/10\/limite-1024x585.jpg","type":"","width":"","height":""}],"author":"giorgio.reveco","twitter_card":"summary_large_image","twitter_title":"\u65e0\u7a77\u6781\u9650\uff1a\u5b9a\u4e49\u4e0e\u793a\u4f8b","twitter_description":"\u4e0d\u61c2\u65e0\u7a77\u6781\u9650\u8ba9\u4f60\u5931\u5206\uff01\u5feb\u6765\u53d1\u73b0\u6700\u4ee4\u4eba\u60ca\u8bb6\u7684\u6280\u5de7\uff0c\u77ac\u95f4\u638c\u63e1\u5b83\u4eec\u3002","twitter_image":"https:\/\/toposuranos.com\/material\/wp-content\/uploads\/2024\/10\/limite.jpg","twitter_creator":"@topuranos","twitter_site":"@topuranos","twitter_misc":{"Escrito por":"giorgio.reveco","Tiempo de lectura":"2 minutos"},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"Article","@id":"https:\/\/toposuranos.com\/material\/zh\/%e6%97%a0%e7%a9%b7%e6%9e%81%e9%99%90%ef%bc%9a%e5%ae%9a%e4%b9%89%e4%b8%8e%e7%a4%ba%e4%be%8b\/#article","isPartOf":{"@id":"https:\/\/toposuranos.com\/material\/zh\/%e6%97%a0%e7%a9%b7%e6%9e%81%e9%99%90%ef%bc%9a%e5%ae%9a%e4%b9%89%e4%b8%8e%e7%a4%ba%e4%be%8b\/"},"author":{"name":"giorgio.reveco","@id":"https:\/\/toposuranos.com\/material\/#\/schema\/person\/e15164361c3f9a2a02cf6c234cf7fdc1"},"headline":"\u65e0\u7a77\u6781\u9650\uff1a\u5b9a\u4e49\u4e0e\u793a\u4f8b","datePublished":"2024-10-22T13:00:33+00:00","dateModified":"2024-10-23T14:36:44+00:00","mainEntityOfPage":{"@id":"https:\/\/toposuranos.com\/material\/zh\/%e6%97%a0%e7%a9%b7%e6%9e%81%e9%99%90%ef%bc%9a%e5%ae%9a%e4%b9%89%e4%b8%8e%e7%a4%ba%e4%be%8b\/"},"wordCount":494,"commentCount":0,"publisher":{"@id":"https:\/\/toposuranos.com\/material\/#organization"},"image":{"@id":"https:\/\/toposuranos.com\/material\/zh\/%e6%97%a0%e7%a9%b7%e6%9e%81%e9%99%90%ef%bc%9a%e5%ae%9a%e4%b9%89%e4%b8%8e%e7%a4%ba%e4%be%8b\/#primaryimage"},"thumbnailUrl":"https:\/\/toposuranos.com\/material\/wp-content\/uploads\/2024\/10\/limite.jpg","articleSection":["\u5fae\u5206","\u6570\u5b66"],"inLanguage":"es","potentialAction":[{"@type":"CommentAction","name":"Comment","target":["https:\/\/toposuranos.com\/material\/zh\/%e6%97%a0%e7%a9%b7%e6%9e%81%e9%99%90%ef%bc%9a%e5%ae%9a%e4%b9%89%e4%b8%8e%e7%a4%ba%e4%be%8b\/#respond"]}]},{"@type":"WebPage","@id":"https:\/\/toposuranos.com\/material\/zh\/%e6%97%a0%e7%a9%b7%e6%9e%81%e9%99%90%ef%bc%9a%e5%ae%9a%e4%b9%89%e4%b8%8e%e7%a4%ba%e4%be%8b\/","url":"https:\/\/toposuranos.com\/material\/zh\/%e6%97%a0%e7%a9%b7%e6%9e%81%e9%99%90%ef%bc%9a%e5%ae%9a%e4%b9%89%e4%b8%8e%e7%a4%ba%e4%be%8b\/","name":"\u65e0\u7a77\u6781\u9650\uff1a\u5b9a\u4e49\u4e0e\u793a\u4f8b - toposuranos.com\/material","isPartOf":{"@id":"https:\/\/toposuranos.com\/material\/#website"},"primaryImageOfPage":{"@id":"https:\/\/toposuranos.com\/material\/zh\/%e6%97%a0%e7%a9%b7%e6%9e%81%e9%99%90%ef%bc%9a%e5%ae%9a%e4%b9%89%e4%b8%8e%e7%a4%ba%e4%be%8b\/#primaryimage"},"image":{"@id":"https:\/\/toposuranos.com\/material\/zh\/%e6%97%a0%e7%a9%b7%e6%9e%81%e9%99%90%ef%bc%9a%e5%ae%9a%e4%b9%89%e4%b8%8e%e7%a4%ba%e4%be%8b\/#primaryimage"},"thumbnailUrl":"https:\/\/toposuranos.com\/material\/wp-content\/uploads\/2024\/10\/limite.jpg","datePublished":"2024-10-22T13:00:33+00:00","dateModified":"2024-10-23T14:36:44+00:00","description":"\u4e0d\u61c2\u65e0\u7a77\u6781\u9650\u8ba9\u4f60\u5931\u5206\uff01\u5feb\u6765\u53d1\u73b0\u6700\u4ee4\u4eba\u60ca\u8bb6\u7684\u6280\u5de7\uff0c\u77ac\u95f4\u638c\u63e1\u5b83\u4eec\u3002","breadcrumb":{"@id":"https:\/\/toposuranos.com\/material\/zh\/%e6%97%a0%e7%a9%b7%e6%9e%81%e9%99%90%ef%bc%9a%e5%ae%9a%e4%b9%89%e4%b8%8e%e7%a4%ba%e4%be%8b\/#breadcrumb"},"inLanguage":"es","potentialAction":[{"@type":"ReadAction","target":["https:\/\/toposuranos.com\/material\/zh\/%e6%97%a0%e7%a9%b7%e6%9e%81%e9%99%90%ef%bc%9a%e5%ae%9a%e4%b9%89%e4%b8%8e%e7%a4%ba%e4%be%8b\/"]}]},{"@type":"ImageObject","inLanguage":"es","@id":"https:\/\/toposuranos.com\/material\/zh\/%e6%97%a0%e7%a9%b7%e6%9e%81%e9%99%90%ef%bc%9a%e5%ae%9a%e4%b9%89%e4%b8%8e%e7%a4%ba%e4%be%8b\/#primaryimage","url":"https:\/\/toposuranos.com\/material\/wp-content\/uploads\/2024\/10\/limite.jpg","contentUrl":"https:\/\/toposuranos.com\/material\/wp-content\/uploads\/2024\/10\/limite.jpg","width":1792,"height":1024},{"@type":"BreadcrumbList","@id":"https:\/\/toposuranos.com\/material\/zh\/%e6%97%a0%e7%a9%b7%e6%9e%81%e9%99%90%ef%bc%9a%e5%ae%9a%e4%b9%89%e4%b8%8e%e7%a4%ba%e4%be%8b\/#breadcrumb","itemListElement":[{"@type":"ListItem","position":1,"name":"Portada","item":"http:\/\/toposuranos.com\/material\/es\/cursos-de-matematica-y-fisica\/"},{"@type":"ListItem","position":2,"name":"\u65e0\u7a77\u6781\u9650\uff1a\u5b9a\u4e49\u4e0e\u793a\u4f8b"}]},{"@type":"WebSite","@id":"https:\/\/toposuranos.com\/material\/#website","url":"https:\/\/toposuranos.com\/material\/","name":"toposuranos.com\/material","description":"","publisher":{"@id":"https:\/\/toposuranos.com\/material\/#organization"},"potentialAction":[{"@type":"SearchAction","target":{"@type":"EntryPoint","urlTemplate":"https:\/\/toposuranos.com\/material\/?s={search_term_string}"},"query-input":{"@type":"PropertyValueSpecification","valueRequired":true,"valueName":"search_term_string"}}],"inLanguage":"es"},{"@type":"Organization","@id":"https:\/\/toposuranos.com\/material\/#organization","name":"toposuranos.com\/material","url":"https:\/\/toposuranos.com\/material\/","logo":{"@type":"ImageObject","inLanguage":"es","@id":"https:\/\/toposuranos.com\/material\/#\/schema\/logo\/image\/","url":"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2023\/10\/logo.png","contentUrl":"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2023\/10\/logo.png","width":2400,"height":2059,"caption":"toposuranos.com\/material"},"image":{"@id":"https:\/\/toposuranos.com\/material\/#\/schema\/logo\/image\/"},"sameAs":["https:\/\/www.facebook.com\/groups\/toposuranos","https:\/\/x.com\/topuranos","https:\/\/www.youtube.com\/channel\/UC16yDm12cPcrwsE0fAM7X1g","https:\/\/www.linkedin.com\/company\/69429190"]},{"@type":"Person","@id":"https:\/\/toposuranos.com\/material\/#\/schema\/person\/e15164361c3f9a2a02cf6c234cf7fdc1","name":"giorgio.reveco","image":{"@type":"ImageObject","inLanguage":"es","@id":"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2023\/10\/1694478625378-96x96.jpeg","url":"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2023\/10\/1694478625378-96x96.jpeg","contentUrl":"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2023\/10\/1694478625378-96x96.jpeg","caption":"giorgio.reveco"},"description":"Soy Licenciado en F\u00edsica, Magister en Ingenier\u00eda Industrial y Docente Universitario. Me dedico a desmitificar la f\u00edsica y las matem\u00e1ticas. Mi objetivo es hacer que estos campos sean f\u00e1cilmente comprensibles para todos, proporcionando las herramientas para explorar no solo el mundo que nos rodea, sino tambi\u00e9n las profundidades de nuestra propia existencia y el orden natural que nos conecta con el cosmos.","sameAs":["http:\/\/toposuranos.com\/material"],"url":"https:\/\/toposuranos.com\/material\/author\/giorgio-reveco\/"}]}},"_links":{"self":[{"href":"https:\/\/toposuranos.com\/material\/wp-json\/wp\/v2\/posts\/29342","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/toposuranos.com\/material\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/toposuranos.com\/material\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/toposuranos.com\/material\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/toposuranos.com\/material\/wp-json\/wp\/v2\/comments?post=29342"}],"version-history":[{"count":0,"href":"https:\/\/toposuranos.com\/material\/wp-json\/wp\/v2\/posts\/29342\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/toposuranos.com\/material\/wp-json\/wp\/v2\/media\/29336"}],"wp:attachment":[{"href":"https:\/\/toposuranos.com\/material\/wp-json\/wp\/v2\/media?parent=29342"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/toposuranos.com\/material\/wp-json\/wp\/v2\/categories?post=29342"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/toposuranos.com\/material\/wp-json\/wp\/v2\/tags?post=29342"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}