{"id":33975,"date":"2024-08-11T13:00:04","date_gmt":"2024-08-11T13:00:04","guid":{"rendered":"https:\/\/toposuranos.com\/material\/?p=33975"},"modified":"2025-08-01T06:02:04","modified_gmt":"2025-08-01T06:02:04","slug":"%e5%ae%9f%e5%a4%89%e6%95%b0%e9%96%a2%e6%95%b0%e3%81%ae%e6%a5%b5%e9%99%90","status":"publish","type":"post","link":"http:\/\/toposuranos.com\/material\/ja\/%e5%ae%9f%e5%a4%89%e6%95%b0%e9%96%a2%e6%95%b0%e3%81%ae%e6%a5%b5%e9%99%90\/","title":{"rendered":"\u5b9f\u5909\u6570\u95a2\u6570\u306e\u6975\u9650"},"content":{"rendered":"<p><center><\/p>\n<h1>\u5b9f\u6570\u5909\u6570\u95a2\u6570\u306e\u6975\u9650<\/h1>\n<p><\/center><\/p>\n<p style=\"text-align:center\"><em><strong>\u8981\u7d04\uff1a<\/strong><br \/>\n\u3053\u306e\u8b1b\u7fa9\u3067\u306f\u3001\u5b9f\u6570\u5909\u6570\u95a2\u6570\u306e\u6975\u9650\u306e\u6b63\u5f0f\u306a\u5b9a\u7fa9\u3092\u6df1\u304f\u691c\u8a0e\u3057\u3001\u305d\u308c\u306b\u57fa\u3065\u3044\u3066\u6975\u9650\u306e\u4ee3\u6570\u306b\u3064\u306a\u304c\u308b\u4e3b\u8981\u306a\u6027\u8cea\u3092\u8a3c\u660e\u3057\u307e\u3059\u3002<\/br><\/em><\/p>\n<p style=\"text-align:center\"><em><strong>\u5b66\u7fd2\u76ee\u6a19\uff1a<\/strong><br \/>\n\u3053\u306e\u8b1b\u7fa9\u306e\u7d42\u4e86\u6642\u306b\u5b66\u751f\u306f\u4ee5\u4e0b\u306e\u3053\u3068\u304c\u3067\u304d\u308b\u3088\u3046\u306b\u306a\u308a\u307e\u3059\uff1a<\/p>\n<ul>\n<li>\u5b9f\u6570\u5909\u6570\u95a2\u6570\u306e\u6975\u9650\u306e<strong>\u5b9a\u7fa9\u3092\u60f3\u8d77\u3059\u308b<\/strong>\u3002<\/li>\n<li><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\epsilon-\\delta<\/span><\/span>\u306b\u3088\u308b\u6f14\u7e79\u3092\u901a\u3058\u3066\u3001\u6975\u9650\u306e\u4ee3\u6570\u306b\u3064\u306a\u304c\u308b\u6027\u8cea\u3092<strong>\u8a3c\u660e\u3059\u308b<\/strong>\u3002<\/li>\n<li>\u6975\u9650\u306e\u4ee3\u6570\u3068\u305d\u306e\u6027\u8cea\u3092\u7528\u3044\u3066\u3001\u5b9f\u6570\u5909\u6570\u95a2\u6570\u306e\u6975\u9650\u3092<strong>\u8a08\u7b97\u3059\u308b<\/strong>\u3002<\/li>\n<\/ul>\n<p><\/em><\/p>\n<p><center><br \/>\n<strong>\u5185\u5bb9\u76ee\u6b21<\/strong><br \/>\n<a href=\"#1\"><strong>\u306f\u3058\u3081\u306b<\/strong><\/a><br \/>\n<a href=\"#2\"><strong>\u95a2\u6570\u306e\u6975\u9650\u306e\u76f4\u611f\u7684\u6982\u5ff5\uff08\u30b0\u30e9\u30d5\u306b\u3088\u308b\u30a2\u30d7\u30ed\u30fc\u30c1\uff09<\/strong><\/a><br \/>\n<a href=\"#3\"><strong>\u6975\u9650\u306e\u6b63\u5f0f\u306a\u5b9a\u7fa9<\/strong><\/a><br \/>\n<a href=\"#4\"><strong>\u6975\u9650\u306e\u6027\u8cea<\/strong><\/a><br \/>\n<a href=\"#5\">\u6975\u9650\u304c\u5b58\u5728\u3059\u308b\u306a\u3089\u3001\u305d\u308c\u306f\u4e00\u610f\u3067\u3042\u308b<\/a><br \/>\n<a href=\"#6\">\u6975\u9650\u306e\u4ee3\u6570<\/a><br \/>\n<a href=\"#7\">\u5358\u7d14\u306a\u6975\u9650\u306e\u8a08\u7b97<\/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>\u306f\u3058\u3081\u306b<\/h2>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=FEPfoAfPsFY&amp;t=240s\" rel=\"noopener\" target=\"_blank\"><strong>\u4ee3\u6570\u5b66\u3084\u5e7e\u4f55\u5b66\u3092\u5b66\u3076\u3053\u3068\u3068\u3001\u5fae\u7a4d\u5206\u3092\u5b66\u3076\u3053\u3068\u3068\u306e\u9055\u3044\u306f\u4f55\u304b\uff1f<\/strong><\/a> \u3053\u306e\u554f\u3044\u306b\u5bfe\u3059\u308b\u7b54\u3048\u306f\u3001\u6975\u9650\u3068\u3044\u3046\u6982\u5ff5\u306b\u3088\u3063\u3066\u660e\u3089\u304b\u306b\u306a\u308a\u307e\u3059\u3002\u672c\u7a3f\u3067\u306f\u6975\u9650\u3068\u305d\u306e\u5b9a\u7fa9\u306b\u3064\u3044\u3066\u5b66\u3073\u307e\u3059\u3002<\/p>\n<p style=\"text-align: justify;\">\u300c\u6975\u9650\u300d\u3068\u3044\u3046\u8a9e\u306f\u901a\u5e38\u3001\u5883\u754c\u306e\u3088\u3046\u306a\u3082\u306e\u3092\u9023\u60f3\u3055\u305b\u307e\u3059\u3002\u305f\u3068\u3048\u3070\u3001\u7aef\u70b9\u304c a, b \u3067\u3042\u308b\u533a\u9593\u306e\u5883\u754c\uff08\u305d\u306e\u6027\u8cea\u306b\u95a2\u308f\u3089\u305a\uff09\u306a\u3069\u3067\u3059\u3002<\/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;\">\u3042\u308b\u3044\u306f\u73fe\u5728\u3068\u3044\u3046\u6642\u9593\u3082\u3001\u904e\u53bb\u3068\u672a\u6765\u306e\u5883\u754c\u3068\u307f\u306a\u3059\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002\u3053\u308c\u3068\u3042\u308b\u7a0b\u5ea6\u4f3c\u305f\u5f62\u3067\u3001\u6975\u9650\u3068\u3044\u3046\u6982\u5ff5\u306f\u3001\u3042\u308b\u70b9\u306b\u6f38\u8fd1\u7684\u306b\u8fd1\u3065\u3044\u3066\u3044\u304f\u3068\u3044\u3046\u76f4\u611f\u7684\u306a\u30a2\u30a4\u30c7\u30a2\u3092\u6570\u5b66\u7684\u306b\u7406\u89e3\u3059\u308b\u305f\u3081\u306e\u67a0\u7d44\u307f\u3092\u63d0\u4f9b\u3057\u307e\u3059\u3002<\/p>\n<p><a name=\"2\"><\/a><\/p>\n<h2>\u30b0\u30e9\u30d5\u7684\u30a2\u30d7\u30ed\u30fc\u30c1\u306b\u3088\u308b\u95a2\u6570\u306e\u6975\u9650\u306e\u76f4\u611f\u7684\u6982\u5ff5<\/h2>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=FEPfoAfPsFY&amp;t=314s\" rel=\"noopener\" target=\"_blank\"><strong>\u6975\u9650\u306e\u6982\u5ff5\u3092\u8996\u899a\u7684\u306b\u7406\u89e3\u3057\u59cb\u3081\u308b\u305f\u3081\u306b\u306f\u3001\u95a2\u6570\u306e\u30b0\u30e9\u30d5\u3092\u63cf\u304f\u3053\u3068\u304b\u3089\u59cb\u3081\u308b\u306e\u304c\u9069\u5207\u3067\u3059<\/strong><\/a>\u3002\u305d\u3057\u3066 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f(x)<\/span><\/span> \u304c\u3001<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x<\/span><\/span> \u304c <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x_0<\/span><\/span> \u306b\u9650\u308a\u306a\u304f\u8fd1\u3065\u304f\u3068\u304d\u306b\u3069\u306e\u3088\u3046\u306a\u6319\u52d5\u3092\u793a\u3059\u304b\u3092\u8003\u5bdf\u3057\u307e\u3059\u3002<\/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=\"l\u00edmite de una funcion\" 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=\"l\u00edmite de una funcion\" class=\"alignnone size-full lazyload\" width=\"692\" height=\"565\" \/><\/noscript><\/center><\/p>\n<p style=\"text-align: justify;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x<\/span><\/span> \u304c <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x_0<\/span><\/span> \u306e\u8fd1\u304f\u306b\u3042\u308b\u306a\u3089\u3070\u3001\u4e2d\u5fc3 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x_0<\/span><\/span>\u3001\u534a\u5f84 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\delta<\/span><\/span> \u306e\u958b\u533a\u9593\u304c\u5b58\u5728\u3057\u3066\u3001<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x<\/span><\/span> \u306f\u305d\u306e\u4e2d\u306b\u542b\u307e\u308c\u307e\u3059\u3002\u3053\u306e\u3053\u3068\u306f\u6b21\u306e3\u901a\u308a\u306e\u65b9\u6cd5\u3067\u8868\u73fe\u3067\u304d\u307e\u3059\uff1a<\/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;\">\u307e\u305f\u306f <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>\u3053\u306e\u6587\u8108\u306b\u304a\u3044\u3066\u306f\u3001\u3053\u308c\u30893\u3064\u306f\u540c\u3058\u5185\u5bb9\u3092\u5225\u306e\u8a00\u3044\u65b9\u3067\u8868\u3057\u3066\u3044\u308b\u306b\u904e\u304e\u307e\u305b\u3093\u3002\u305f\u3060\u3057\u6700\u5f8c\u306e\u8868\u73fe\uff08\u4e2d\u5fc3 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x_0<\/span><\/span>\u3001\u534a\u5f84 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\delta<\/span><\/span> \u306e\u958b\u7403\u306b\u5c5e\u3059\u308b\uff09\u306f\u3001<strong>\u4f4d\u76f8\u7a7a\u9593\u8ad6\u306e\u8b1b\u7fa9<\/strong>\u306a\u3069\u3067\u300c\u8fd1\u63a5\u6027\u300d\u3068\u3044\u3046\u6982\u5ff5\u3092\u3088\u308a\u6df1\u304f\u6271\u3046\u969b\u306b\u9069\u3057\u3066\u3044\u307e\u3059\u3002<\/em><\/p>\n<p style=\"text-align: justify;\">\u3053\u306e\u3088\u3046\u306a\u72b6\u6cc1\u4e0b\u3067\u306f\u3001\u4e2d\u5fc3 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">l<\/span><\/span>\u3001\u534a\u5f84 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\epsilon<\/span><\/span> \u306e\u958b\u533a\u9593\u304c\u5b58\u5728\u3057\u3001<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f(x)<\/span><\/span> \u304c\u305d\u306e\u4e2d\u306b\u542b\u307e\u308c\u308b\u3001\u3059\u306a\u308f\u3061\uff1a<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">|f(x) - l|\\lt \\epsilon<\/span><\/span> \u304c\u6210\u308a\u7acb\u3061\u307e\u3059\u3002<\/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=\"l\u00edmite de una funcion\" 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=\"l\u00edmite de una funcion\" class=\"alignnone size-full lazyload\" width=\"625\" height=\"549\" \/><\/noscript><\/center><\/p>\n<p style=\"text-align: justify;\">\u3053\u3053\u306b\u3001\u6975\u9650\u306e\u6570\u5b66\u7684\u6982\u5ff5\u306e\u6839\u5e79\u304c\u73fe\u308c\u307e\u3059\u3002\u305d\u308c\u306f\u4ee5\u4e0b\u306e\u6761\u4ef6\u304c\u6210\u308a\u7acb\u3064\u3068\u304d\u306b\u6975\u9650\u304c\u5b58\u5728\u3059\u308b\u3068\u3044\u3046\u3053\u3068\u3067\u3059\uff1a<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0 \\lt|x-x_0|\\lt \\delta<\/span><\/span> \u306a\u3089\u3070 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">|f(x)-l|\\lt \\epsilon<\/span><\/span>\u3002\u3053\u306e\u3068\u304d\u3001<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">l<\/span><\/span> \u306f <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x<\/span><\/span> \u304c <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x_0<\/span><\/span> \u306b\u9650\u308a\u306a\u304f\u8fd1\u3065\u304f\u3068\u304d\u306e\u95a2\u6570\u306e\u6975\u9650\u5024\u306b\u306a\u308a\u307e\u3059\u3002<\/p>\n<p><a name=\"3\"><\/a><\/p>\n<h2>\u6975\u9650\u306e\u6b63\u5f0f\u306a\u5b9a\u7fa9<\/h2>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=FEPfoAfPsFY&amp;t=689s\" rel=\"noopener\" target=\"_blank\"><strong>\u4e0a\u3067\u63d0\u793a\u3055\u308c\u305f\u76f4\u611f\u7684\u304b\u3064\u56f3\u7684\u306a\u7406\u89e3\u304b\u3089\u3001\u6975\u9650\u306e\u6b63\u5f0f\u306a\u5b9a\u7fa9\u3092\u660e\u78ba\u306b\u3059\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002<\/strong> <\/a>\u6975\u9650\u304c\u5b58\u5728\u3059\u308b\u3068\u306f\u3001\u4efb\u610f\u306e <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\epsilon<\/span><\/span>\uff08\u3059\u306a\u308f\u3061\u3001<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f(x)<\/span><\/span> \u3068 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">l<\/span><\/span> \u3068\u306e\u8ddd\u96e2\uff09\u306b\u5bfe\u3057\u3066\u3001\u3042\u308b <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\delta<\/span><\/span> \u304c\u5b58\u5728\u3057\u3066\u3001<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0 \\lt|x-x_0|\\lt \\delta<\/span><\/span> \u306a\u3089\u3070 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">|f(x) - l|\\lt \\epsilon<\/span><\/span> \u304c\u6210\u308a\u7acb\u3064\u3068\u3044\u3046\u3053\u3068\u3067\u3059\u3002\u3053\u306e\u6982\u5ff5\u306f\u6700\u521d\u306f\u6349\u3048\u306b\u304f\u304f\u3001\u4e16\u754c\u4e2d\u306e\u591a\u304f\u306e\u5fae\u7a4d\u5206\u306e\u5b66\u751f\u306b\u3068\u3063\u3066\u6d99\u306e\u7a2e\u3068\u306a\u308b\u3082\u306e\u3067\u3059\u304c\u3001\u6b21\u306e\u8868\u73fe\u3067\u8981\u7d04\u3059\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3059\uff1a<\/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>\u6975\u9650\u306e\u6027\u8cea<\/h2>\n<p style=\"text-align: justify;\">\u6975\u9650\u306e\u6b63\u5f0f\u306a\u5b9a\u7fa9\u3092\u6301\u3064\u3053\u3068\u306e\u91cd\u8981\u6027\u306f\u3001\u305d\u3053\u304b\u3089\u76f4\u611f\u7684\u306a\u3082\u306e\u3082\u305d\u3046\u3067\u306a\u3044\u3082\u306e\u3082\u542b\u3081\u3066\u3001\u6975\u9650\u306b\u95a2\u3059\u308b\u6027\u8cea\u3092\u8a3c\u660e\u3067\u304d\u308b\u3068\u3044\u3046\u70b9\u306b\u3042\u308a\u307e\u3059\u3002<\/p>\n<p style=\"text-align: justify;\">\u3053\u3053\u304b\u3089\u5148\u3078\u9032\u3080\u524d\u306b\u3001\u5fc5\u9808\u3067\u306f\u3042\u308a\u307e\u305b\u3093\u304c\u3001\u4ee5\u4e0b\u306e\u30ea\u30f3\u30af\u5148\u306b\u3042\u308b <a href=\"http:\/\/toposuranos.com\/material\/es\/category\/matematica\/logica-matematica\/logica-proposicional\/\" rel=\"noopener\" target=\"_blank\"><strong>\u6570\u5b66\u8ad6\u7406<\/strong><\/a> \u306e\u3044\u304f\u3064\u304b\u306e\u57fa\u672c\u6982\u5ff5\u3092\u5fa9\u7fd2\u3057\u3066\u304a\u304f\u3053\u3068\u3092\u5f37\u304f\u63a8\u5968\u3057\u307e\u3059\u3002\u305d\u3046\u3059\u308b\u3053\u3068\u3067\u3001\u3053\u308c\u304b\u3089\u51fa\u3066\u304f\u308b\u8a3c\u660e\u3092\u3088\u308a\u5bb9\u6613\u306b\u7406\u89e3\u3067\u304d\u308b\u3067\u3057\u3087\u3046\u3002<\/p>\n<p><a name=\"5\"><\/a><\/p>\n<h3>\u6975\u9650\u304c\u5b58\u5728\u3059\u308b\u306a\u3089\u3070\u3001\u305d\u308c\u306f\u4e00\u610f\u3067\u3042\u308b<\/h3>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=FEPfoAfPsFY&amp;t=904s\" rel=\"noopener\" target=\"_blank\"><strong>\u3053\u306e\u6027\u8cea\u3092\u8a3c\u660e\u3059\u308b\u305f\u3081\u306b\u3001\u80cc\u7406\u6cd5\uff08\u5e30\u8b2c\u6cd5\uff09\u3092\u7528\u3044\u307e\u3059\u3002<\/strong><\/a> \u307e\u305a\u3001\u6b21\u306e\u524d\u63d0\u96c6\u5408\u3092\u5b9a\u7fa9\u3059\u308b\u3068\u3053\u308d\u304b\u3089\u59cb\u3081\u307e\u3059\uff1a<\/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;\">\u3053\u308c\u3089\u306e\u524d\u63d0\u306b\u57fa\u3065\u3044\u3066\u3001\u6b21\u306e\u5f62\u5f0f\u7684\u306a\u8a3c\u660e\u3092\u69cb\u7bc9\u3067\u304d\u307e\u3059\uff1a<\/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>\u4eee\u5b9a<\/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>; \u4eee\u5b9a<\/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>; \u4eee\u5b9a<\/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>\u5c0e\u5165<\/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>\u5358\u8abf\u6027<\/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>; \u306a\u305c\u306a\u3089 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">L \\lt L^\\prime <\/span><\/span> \u3060\u304b\u3089<\/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>; (5,6) \u3092\u4f7f\u7528<\/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><br \/>\n<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><br \/>\n<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\wedge<\/span><\/span><br \/>\n<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><br \/>\n<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><br \/>\n<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><br \/>\n<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\wedge<\/span><\/span><br \/>\n<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><br \/>\n<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><br \/>\n<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><br \/>\n<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\wedge<\/span><\/span><br \/>\n<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><br \/>\n<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>; (1,2,6,7) \u306b\u3088\u308b<\/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>; (8) \u3068\u540c\u69d8\u306e\u624b\u9806<\/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>-\u5c0e\u5165(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>; \u5b9a\u7fa9(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>\u30e2\u30fc\u30c0\u30b9\u30dd\u30cd\u30f3\u30b9<\/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>\u80cc\u7406\u6cd5<\/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;\">\u3053\u306e\u8a3c\u660e\u304b\u3089\u5c0e\u304b\u308c\u308b\u306e\u306f\u3001\u3082\u30572\u3064\u306e\u6975\u9650\u304c\u5b58\u5728\u3059\u308b\u306a\u3089\u3070\u3001\u305d\u308c\u3089\u306f\u7b49\u3057\u304f\u306a\u3051\u308c\u3070\u306a\u3089\u305a\u3001\u3057\u305f\u304c\u3063\u3066\u6975\u9650\u306f\u4e00\u610f\u3067\u3042\u308b\u3068\u3044\u3046\u3053\u3068\u3067\u3059\u3002<\/p>\n<p><a name=\"6\"><\/a><\/p>\n<h3>\u6975\u9650\u306e\u4ee3\u6570<\/h3>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=FEPfoAfPsFY&amp;t=2011s\" rel=\"noopener\" target=\"_blank\"><strong>\u3053\u3053\u307e\u3067\u3067\u3001\u6975\u9650\u3068\u3044\u3046\u6570\u5b66\u7684\u6982\u5ff5\u306e\u672c\u8cea\u306b\u3064\u3044\u3066\u898b\u3066\u304d\u307e\u3057\u305f\u3002<\/strong><\/a> \u3057\u304b\u3057\u3001\u3053\u308c\u3060\u3051\u3067\u306f\u6975\u9650\u3092\u5b9f\u969b\u306b\u8a08\u7b97\u3059\u308b\u306b\u306f\u4e0d\u5341\u5206\u3067\u3059\u3002\u6975\u9650\u306e\u5b9a\u7fa9\u3092\u4f7f\u3063\u3066\u8a08\u7b97\u3059\u308b\u306a\u3069\u3068\u3044\u3046\u306e\u306f\u3001\u82e6\u75db\u3092\u6c42\u3081\u308b\u72c2\u4eba\u306e\u3059\u308b\u3053\u3068\u3067\u3059\u3002\u3053\u306e\u554f\u984c\u3092\u89e3\u6c7a\u3059\u308b\u305f\u3081\u306b\u3001\u4eca\u5f8c\u306f\u5b9f\u969b\u306b\u6975\u9650\u3092\u8a08\u7b97\u3067\u304d\u308b\u3088\u3046\u306b\u3059\u308b\u305f\u3081\u306e\u30c6\u30af\u30cb\u30c3\u30af\u3092\u6271\u3063\u3066\u3044\u304d\u307e\u3059\u3002<\/p>\n<p style=\"text-align: justify;\">\u3053\u3053\u3067\u3001<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x_0, \\alpha, \\beta, L, M \\in \\mathbb{R},<\/span><\/span> \u304b\u3064 f \u304a\u3088\u3073 g \u3092\u6b21\u306e\u6761\u4ef6\u3092\u6e80\u305f\u3059\u5b9f\u6570\u5024\u95a2\u6570\u3068\u3057\u307e\u3059\uff1a<\/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;\">\u3053\u306e\u3068\u304d\u3001\u6b21\u306e\u6027\u8cea\u304c\u6210\u308a\u7acb\u3061\u307e\u3059\uff1a<\/p>\n<h4>\u95a2\u6570\u306e\u548c\u304a\u3088\u3073\u5dee\u306e\u6975\u9650<\/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>\u8a3c\u660e\uff1a<\/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>\u6b21\u306e\u524d\u63d0\u96c6\u5408\u3092\u8003\u3048\u307e\u3059<\/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>\u3002\u3053\u306e\u524d\u63d0\u306b\u57fa\u3065\u304d\u3001\u6b21\u306e\u3088\u3046\u306b\u63a8\u8ad6\u3092\u9032\u3081\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3059\uff1a<\/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>; \u4eee\u5b9a<\/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>; \u5b9a\u7fa9<\/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) \u3088\u308a<\/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>; \u4eee\u5b9a<\/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) \u306b\u985e\u4f3c<\/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) \u3088\u308a<\/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>; \u4e09\u89d2\u4e0d\u7b49\u5f0f\uff1a<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) \u3088\u308a<\/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>; \u5b9a\u7fa9<\/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) \u3088\u308a<\/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>; \u5b9a\u7fa9<\/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) \u306b\u985e\u4f3c<\/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) \u3088\u308a<\/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) \u3088\u308a<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h4>\u95a2\u6570\u306e\u7a4d\u306e\u6975\u9650<\/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>\u3053\u306e\u8a3c\u660e\u306f\u524d\u306e\u3082\u306e\u3088\u308a\u3084\u3084\u96e3\u3057\u3044\u3067\u3059\u304c\u3001<\/strong><\/a> \u3044\u304f\u3064\u304b\u306e\u201c\u30c9\u30e9\u30b3\u30cb\u30c3\u30af\u306a\u201d\u30c6\u30af\u30cb\u30c3\u30af\u3092\u7528\u3044\u308c\u3070\u89e3\u6c7a\u3067\u304d\u307e\u3059\u3002\u524d\u3068\u540c\u3058\u524d\u63d0\u96c6\u5408 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathcal{H}<\/span><\/span> \u3092\u7528\u3044\u3066\u3001\u6b21\u306e\u3088\u3046\u306a\u63a8\u8ad6\u3092\u69cb\u7bc9\u3067\u304d\u307e\u3059\uff1a<\/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>; \u5b9a\u7fa9<\/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>; \u4eee\u5b9a<\/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) \u3092\u4f7f\u7528<\/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>; \u5b9a\u7fa9<\/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>; \u4eee\u5b9a<\/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) \u3092\u4f7f\u7528<\/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><br \/>\n<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>; \u4e09\u89d2\u4e0d\u7b49\u5f0f + \u7279\u5225\u306a\u5834\u5408 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\overline{\\epsilon}<\/span><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: right;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(6)<\/span><\/span><\/td>\n<td style=\"text-align: left;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\mathcal{H}\\vdash |f(x)|\\lt 1 + |L| <\/span><\/span>; (5) \u3088\u308a<\/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>; \u4e09\u89d2\u4e0d\u7b49\u5f0f + \u7279\u5225\u306a\u5834\u5408 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\overline{\\overline{\\epsilon}}<\/span><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: right;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(8)<\/span><\/span><\/td>\n<td style=\"text-align: left;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\mathcal{H}\\vdash |g(x)| \\lt 1 + |M| <\/span><\/span>; (7) \u3088\u308a<\/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><br \/>\n<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><br \/>\n<span style=\"background-color: #a0a0ff; color:#000000\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">+ Mf(x) - LM<\/span><\/span><\/span><br \/>\n<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">|<\/span><\/span>; \u30bc\u30ed\u3092\u52a0\u3048\u308b\u64cd\u4f5c<\/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><br \/>\n<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><br \/>\n<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><br \/>\n<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">|<\/span><\/span>; \u56e0\u6570\u5206\u89e3<\/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><br \/>\n<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><br \/>\n<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><br \/>\n<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">|<\/span><\/span>; \u4e09\u89d2\u4e0d\u7b49\u5f0f (9) \u3088\u308a<\/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><br \/>\n<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><br \/>\n<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><br \/>\n<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><br \/>\n<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) \u3088\u308a<\/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) \u3088\u308a<\/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) \u3088\u308a<\/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) \u3088\u308a<\/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) \u3088\u308a<\/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>\u5b9a\u6570\u95a2\u6570\u306e\u6975\u9650<\/h4>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=FEPfoAfPsFY&amp;t=3450s\" rel=\"noopener\" target=\"_blank\"><strong>\u5b9a\u6570\u95a2\u6570\u306e\u6975\u9650<\/strong><\/a> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f(x)=c<\/span><\/span> \u306f\u3001\u5b9a\u6570 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">c<\/span><\/span> \u81ea\u8eab\u3067\u3059\u3002\u3059\u306a\u308f\u3061\uff1a<\/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>\u8a3c\u660e<\/strong><\/p>\n<p style=\"text-align: justify;\">\u3053\u306e\u8a3c\u660e\u306f\u5b9f\u969b\u306b\u306f\u975e\u5e38\u306b\u7c21\u5358\u3067\u3001\u4e8b\u5b9f\u4e0a\u30c8\u30fc\u30c8\u30ed\u30b8\u30fc\u3067\u3059\u3002\u4ee5\u4e0b\u304c\u3059\u3067\u306b\u77e5\u3089\u308c\u3066\u3044\u307e\u3059\uff1a<\/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;\">\u3057\u304b\u3057 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0=|c-c|\\lt \\epsilon<\/span><\/span> \u306f\u4efb\u610f\u306e\u6b63\u306e \u03b5 \u306b\u5bfe\u3057\u3066\u5e38\u306b\u771f\u3067\u3042\u308b\u305f\u3081\u3001\u542b\u610f\u3082\u307e\u305f\u30c8\u30fc\u30c8\u30ed\u30b8\u30fc\u3068\u306a\u308a\u3001\u3057\u305f\u304c\u3063\u3066 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to x_0}c = c <\/span><\/span> \u3082\u307e\u305f\u30c8\u30fc\u30c8\u30ed\u30b8\u30fc\u3067\u3042\u308b\u3053\u3068\u304c\u308f\u304b\u308a\u307e\u3059\u3002<\/p>\n<h4>\u95a2\u6570\u306e\u5546\u306e\u6975\u9650<\/h4>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=FEPfoAfPsFY&amp;t=3563s\" rel=\"noopener\" target=\"_blank\"><strong>\u6b21\u306b\u30012\u3064\u306e\u95a2\u6570\u306e\u5546\u306e\u6975\u9650\u306b\u95a2\u3059\u308b\u898f\u5247\u3092\u8a3c\u660e\u3057\u307e\u3059\u3002<\/strong><\/a> \u305d\u308c\u306f\u6b21\u306e\u901a\u308a\u3067\u3059\uff1a<\/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;\">\u3053\u308c\u307e\u3067\u3068\u540c\u69d8\u306b\u3001\u6b21\u306e\u524d\u63d0\u96c6\u5408\u304c\u6210\u308a\u7acb\u3064\u3068\u4eee\u5b9a\u3057\u307e\u3059\uff1a<\/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>\u8a3c\u660e<\/strong><\/p>\n<p style=\"text-align: justify;\">\u5e78\u3044\u306b\u3082\u3001\u3053\u308c\u307e\u3067\u884c\u3063\u3066\u304d\u305f\u3088\u3046\u306a\u8907\u96d1\u306a\u8a3c\u660e\u3092\u7e70\u308a\u8fd4\u3059\u5fc5\u8981\u306f\u3042\u308a\u307e\u305b\u3093\u3002\u3053\u308c\u4ee5\u964d\u306f\u3001\u3059\u3067\u306b\u8a3c\u660e\u6e08\u307f\u306e\u6027\u8cea\u3092\u6d3b\u7528\u3059\u308b\u3053\u3068\u3067\u76ee\u7684\u3092\u9054\u6210\u3067\u304d\u307e\u3059\u3002\u305f\u3060\u3057\u305d\u306e\u524d\u306b\u3001\u307e\u305a\u6b21\u306e\u3053\u3068\u3092\u8a3c\u660e\u3057\u3066\u304a\u304d\u307e\u3059\uff1a<\/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;\">\u3053\u308c\u3092\u8a3c\u660e\u3059\u308b\u306b\u306f\u3001\u5b9a\u6570\u95a2\u6570\u306e\u6975\u9650\u3068\u7a4d\u306e\u6975\u9650\u306e\u6cd5\u5247\u3092\u7d44\u307f\u5408\u308f\u305b\u3066\u7528\u3044\u308c\u3070\u5341\u5206\u3067\u3059\u3002\u305f\u3060\u3057\u3001<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">g(x)<\/span><\/span> \u304c\u30bc\u30ed\u3067\u306a\u3044\u3053\u3068\u306b\u6ce8\u610f\u3059\u308b\u5fc5\u8981\u304c\u3042\u308a\u307e\u3059\uff1a<\/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;\">\u3088\u3063\u3066\u3001<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;\">\u6700\u5f8c\u306b\u3001\u7a4d\u306e\u6975\u9650\u306e\u6cd5\u5247\u3088\u308a\u6b21\u304c\u6210\u308a\u7acb\u3061\u307e\u3059\uff1a<\/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;\">\u3053\u306e\u7d50\u679c\u306f\u3001<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">M<\/span><\/span> \u304c\u30bc\u30ed\u3067\u306a\u3044\u5834\u5408\u306b\u9650\u308a\u6210\u308a\u7acb\u3061\u307e\u3059\u3002<\/p>\n<h4>\u81ea\u7136\u6570\u4e57\u306e\u6975\u9650<\/h4>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=FEPfoAfPsFY&amp;t=3725s\" rel=\"noopener\" target=\"_blank\"><strong>\u3053\u306e\u6027\u8cea\u306f\u6b21\u306e\u3053\u3068\u3092\u4e3b\u5f35\u3057\u3066\u3044\u307e\u3059\uff1a<\/strong><\/a><br \/>\n<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x_0 \\to x_0}f(x) = L<\/span><\/span> \u306e\u3068\u304d\u3001\u6b21\u304c\u6210\u308a\u7acb\u3061\u307e\u3059\uff1a<br \/>\n<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>\u3002\u3053\u308c\u306f\u6570\u5b66\u7684\u5e30\u7d0d\u6cd5\u306b\u3088\u308a\u8a3c\u660e\u3067\u304d\u307e\u3059\u3002<\/p>\n<p style=\"text-align: justify; color: #000080;\"><strong>\u8a3c\u660e\uff1a<\/strong><\/p>\n<ul style=\"text-align: justify;\">\n<li><strong>\u5834\u5408 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">n=1<\/span><\/span>\uff1a<\/strong>\uff08\u521d\u671f\u30b9\u30c6\u30c3\u30d7\uff09\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> \u3088\u3063\u3066\u521d\u671f\u30b9\u30c6\u30c3\u30d7\u306f\u5b8c\u4e86 \u2705<\/p>\n<\/li>\n<li><strong>\u5834\u5408 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">n=k<\/span><\/span>\uff1a<\/strong>\uff08\u5e30\u7d0d\u30b9\u30c6\u30c3\u30d7\uff09\n<p style=\"text-align: justify;\">\u6b21\u306e\u4eee\u5b9a\u3092\u7f6e\u304d\u307e\u3059\uff1a<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to x_0} [f(x)]^k = L^k <\/span><\/span>\uff08\u5e30\u7d0d\u4eee\u5b9a\uff09\u3002\u3053\u306e\u3068\u304d\u3001\u4ee5\u4e0b\u3092\u793a\u3057\u307e\u3059\uff1a<br \/>\n<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><\/p>\n<p style=\"text-align: justify;\">\u78ba\u304b\u306b\u3001<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>\u3002\u3053\u308c\u306f\u4e0a\u3067\u793a\u3057\u305f\u7a4d\u306e\u6975\u9650\u306e\u6027\u8cea\u306b\u3088\u308a\u6210\u308a\u7acb\u3061\u307e\u3059\u3002<\/p>\n<p style=\"text-align: justify;\">\u3057\u305f\u304c\u3063\u3066\u3001\u5e30\u7d0d\u4eee\u5b9a\u3088\u308a\uff1a<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> \u3088\u3063\u3066\u5e30\u7d0d\u30b9\u30c6\u30c3\u30d7\u3082\u5b8c\u4e86 \u2705<\/p>\n<\/li>\n<li>\u3086\u3048\u306b\uff1a<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 \u4e57\u6839\u306e\u6975\u9650<\/h4>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=FEPfoAfPsFY&amp;t=3912s\" rel=\"noopener\" target=\"_blank\"><strong>\u7d2f\u4e57\u306e\u5834\u5408\u3068\u540c\u69d8\u306b\u3001\u6b21\u304c\u6210\u308a\u7acb\u3061\u307e\u3059\uff1a<\/strong><\/a><br \/>\n<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>\u8a3c\u660e\uff1a<\/strong><\/p>\n<p style=\"text-align: justify;\">\u5148\u307b\u3069\u8a3c\u660e\u3057\u305f\u7d2f\u4e57\u306e\u6975\u9650\u6cd5\u5247\u3092\u4f7f\u3048\u3070\u3001\u6b21\u306e\u3088\u3046\u306b\u66f8\u3051\u307e\u3059\uff1a<\/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;\">\u3057\u305f\u304c\u3063\u3066\uff1a<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>\u5206\u6570\u3079\u304d\u306e\u6975\u9650<\/h4>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=FEPfoAfPsFY&amp;t=4007s\" rel=\"noopener\" target=\"_blank\"><strong>\u524d\u306e2\u3064\u306e\u8a3c\u660e\u3092\u5408\u308f\u305b\u3066\u3001<\/strong><\/a> \u6700\u5f8c\u306e\u8a3c\u660e\u3092\u5c0e\u304f\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002\u305d\u308c\u306f\uff1a<br \/>\n<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><br \/>\n\u3053\u308c\u306f\u7a4d\u306e\u6975\u9650\u6cd5\u5247\u304b\u3089\u5c0e\u304b\u308c\u307e\u3059\u3002\u306a\u305c\u306a\u3089\u3001<br \/>\n<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle [f(x)]^{\\frac{p}{q}} =[\\sqrt[q]{f(x)}]^p <\/span><\/span> \u304a\u3088\u3073<br \/>\n<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle L^{\\frac{p}{q}} =[\\sqrt[q]{L}]^p. <\/span><\/span><br \/>\n\u3067\u3042\u308b\u304b\u3089\u3067\u3059\u3002<\/p>\n<h4>\u6975\u9650 <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>\u3053\u306e\u8a3c\u660e\u3092\u3082\u3063\u3066\u3001\u4e00\u9023\u306e\u8a3c\u660e\u3092\u7de0\u3081\u304f\u304f\u308a\u307e\u3059\u3002<\/strong><\/a><br \/>\n\u3053\u308c\u3068\u5148\u884c\u3059\u308b\u6027\u8cea\u3092\u4f7f\u3048\u3070\u3001\u4eca\u5f8c\u306f\u591a\u304f\u306e\u6975\u9650\u3092\u76f4\u611f\u7684\u306b\u8a08\u7b97\u3059\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002<\/p>\n<p style=\"text-align: justify;\">\u6b21\u3092\u793a\u3059\u306e\u306f\u7c21\u5358\u3067\u3059\uff1a<br \/>\n<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to x_0}x = x_0<\/span><\/span>\u3002<br \/>\n\u3053\u306e\u305f\u3081\u306b\u306f\u3001\u6b21\u306e\u6761\u4ef6\u3092\u6e80\u305f\u3059\u5fc5\u8981\u304c\u3042\u308a\u307e\u3059\uff1a<\/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;\">\u6975\u9650\u306e\u5b9a\u7fa9\u306b\u3088\u308c\u3070\u3001\u4efb\u610f\u306e \u03b5 \u306b\u5bfe\u3057\u3066\u3001\u5c11\u306a\u304f\u3068\u3082\u4e00\u3064\u306e \u03b4 \u304c\u5b58\u5728\u3059\u308c\u3070\u6761\u4ef6\u304c\u6210\u308a\u7acb\u3061\u307e\u3059\u3002\u3057\u305f\u304c\u3063\u3066\u3001\u03b4 \u3092\u4e00\u3064\u898b\u3064\u3051\u308b\u3060\u3051\u3067\u5341\u5206\u3067\u3059\u3002\u5b9f\u969b\u306b\u306f\u3001<br \/>\n<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\delta\\leq\\epsilon<\/span><\/span> \u3067\u3042\u308b\u4efb\u610f\u306e \u03b4 \u304c\u3053\u306e\u6761\u4ef6\u3092\u6e80\u305f\u3059\u3053\u3068\u304c\u308f\u304b\u308a\u307e\u3059\u3002<br \/>\n\u3088\u3063\u3066\uff1a<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>\u7c21\u5358\u306a\u6975\u9650\u306e\u8a08\u7b97<\/h2>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=FEPfoAfPsFY&amp;t=4155s\" rel=\"noopener\" target=\"_blank\"><strong>\u3053\u308c\u307e\u3067\u898b\u3066\u304d\u305f\u3059\u3079\u3066\u306e\u5b9a\u7406\u306e\u304a\u304b\u3052\u3067\u3001<\/strong><\/a> \u591a\u304f\u306e\u6975\u9650\u3092\u975e\u5e38\u306b\u76f4\u611f\u7684\u306b\u8a08\u7b97\u3059\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002\u307e\u308b\u3067\u95a2\u6570\u3092\u305d\u306e\u307e\u307e\u8a55\u4fa1\u3059\u308b\u304b\u306e\u3088\u3046\u306b\u6271\u3048\u307e\u3059\u3002\u4ee5\u4e0b\u306b\u3044\u304f\u3064\u304b\u306e\u4f8b\u3092\u793a\u3057\u307e\u3059\uff1a<\/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^2 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>\u5b9f\u6570\u5909\u6570\u95a2\u6570\u306e\u6975\u9650 \u8981\u7d04\uff1a \u3053\u306e\u8b1b\u7fa9\u3067\u306f\u3001\u5b9f\u6570\u5909\u6570\u95a2\u6570\u306e\u6975\u9650\u306e\u6b63\u5f0f\u306a\u5b9a\u7fa9\u3092\u6df1\u304f\u691c\u8a0e\u3057\u3001\u305d\u308c\u306b\u57fa\u3065\u3044\u3066\u6975\u9650\u306e\u4ee3\u6570\u306b\u3064\u306a\u304c\u308b\u4e3b\u8981\u306a\u6027\u8cea\u3092\u8a3c\u660e\u3057\u307e\u3059\u3002 \u5b66\u7fd2\u76ee\u6a19\uff1a \u3053\u306e\u8b1b\u7fa9\u306e\u7d42\u4e86\u6642\u306b\u5b66\u751f\u306f\u4ee5\u4e0b\u306e\u3053\u3068\u304c\u3067\u304d\u308b\u3088\u3046\u306b\u306a\u308a\u307e\u3059\uff1a \u5b9f\u6570\u5909\u6570\u95a2\u6570\u306e\u6975\u9650\u306e\u5b9a\u7fa9\u3092\u60f3\u8d77\u3059\u308b\u3002 \u306b\u3088\u308b\u6f14\u7e79\u3092\u901a\u3058\u3066\u3001\u6975\u9650\u306e\u4ee3\u6570\u306b\u3064\u306a\u304c\u308b\u6027\u8cea\u3092\u8a3c\u660e\u3059\u308b\u3002 \u6975\u9650\u306e\u4ee3\u6570\u3068\u305d\u306e\u6027\u8cea\u3092\u7528\u3044\u3066\u3001\u5b9f\u6570\u5909\u6570\u95a2\u6570\u306e\u6975\u9650\u3092\u8a08\u7b97\u3059\u308b\u3002 \u5185\u5bb9\u76ee\u6b21 \u306f\u3058\u3081\u306b \u95a2\u6570\u306e\u6975\u9650\u306e\u76f4\u611f\u7684\u6982\u5ff5\uff08\u30b0\u30e9\u30d5\u306b\u3088\u308b\u30a2\u30d7\u30ed\u30fc\u30c1\uff09 \u6975\u9650\u306e\u6b63\u5f0f\u306a\u5b9a\u7fa9 \u6975\u9650\u306e\u6027\u8cea \u6975\u9650\u304c\u5b58\u5728\u3059\u308b\u306a\u3089\u3001\u305d\u308c\u306f\u4e00\u610f\u3067\u3042\u308b \u6975\u9650\u306e\u4ee3\u6570 \u5358\u7d14\u306a\u6975\u9650\u306e\u8a08\u7b97 \u306f\u3058\u3081\u306b \u4ee3\u6570\u5b66\u3084\u5e7e\u4f55\u5b66\u3092\u5b66\u3076\u3053\u3068\u3068\u3001\u5fae\u7a4d\u5206\u3092\u5b66\u3076\u3053\u3068\u3068\u306e\u9055\u3044\u306f\u4f55\u304b\uff1f \u3053\u306e\u554f\u3044\u306b\u5bfe\u3059\u308b\u7b54\u3048\u306f\u3001\u6975\u9650\u3068\u3044\u3046\u6982\u5ff5\u306b\u3088\u3063\u3066\u660e\u3089\u304b\u306b\u306a\u308a\u307e\u3059\u3002\u672c\u7a3f\u3067\u306f\u6975\u9650\u3068\u305d\u306e\u5b9a\u7fa9\u306b\u3064\u3044\u3066\u5b66\u3073\u307e\u3059\u3002 \u300c\u6975\u9650\u300d\u3068\u3044\u3046\u8a9e\u306f\u901a\u5e38\u3001\u5883\u754c\u306e\u3088\u3046\u306a\u3082\u306e\u3092\u9023\u60f3\u3055\u305b\u307e\u3059\u3002\u305f\u3068\u3048\u3070\u3001\u7aef\u70b9\u304c a, b \u3067\u3042\u308b\u533a\u9593\u306e\u5883\u754c\uff08\u305d\u306e\u6027\u8cea\u306b\u95a2\u308f\u3089\u305a\uff09\u306a\u3069\u3067\u3059\u3002 , 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