{"id":2139,"date":"2024-05-17T21:50:13","date_gmt":"2024-05-17T12:50:13","guid":{"rendered":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/?page_id=2139"},"modified":"2024-05-17T21:50:10","modified_gmt":"2024-05-17T12:50:10","slug":"%e3%83%86%e3%82%a4%e3%83%a9%e3%83%bc%e5%b1%95%e9%96%8b","status":"publish","type":"page","link":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/%e7%90%86%e5%b7%a5%e7%b3%bb%e3%81%ae%e6%95%b0%e5%ad%a6b\/%e3%83%86%e3%82%a4%e3%83%a9%e3%83%bc%e5%b1%95%e9%96%8b\/","title":{"rendered":"\u30c6\u30a4\u30e9\u30fc\u5c55\u958b"},"content":{"rendered":"<p>\\(f(x)\\) \u304c \\(x = a\\) \u3092\u542b\u3080\u533a\u9593\u3067\u5fae\u5206\u53ef\u80fd\u3067\u3042\u308b\u3068\u304d\uff0c\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u5c55\u958b\u3067\u304d\u308b\u3002\u3053\u308c\u3092\u300c\\(x = a\\) \u306e\u307e\u308f\u308a\u306e<span style=\"font-family: helvetica, arial, sans-serif;\"><strong>\u30c6\u30a4\u30e9\u30fc\u5c55\u958b<\/strong><\/span>\u300d\u3068\u547c\u3076\u3002<br \/>\n\\begin{eqnarray}<br \/>\nf(a + x) &amp;=&amp; f(a) + f'(a)\\, x + \\frac{f^{&#8221;}(a)}{2!}\\, x^2 + \\cdots +<br \/>\n\\frac{f^{({n})}(a)}{n!}\\, x^n + \\cdots\\\\<br \/>\n&amp;=&amp; f(a) + \\sum_{k=1} \\frac{f^{(k)}(a)}{k!}\\, x^k<br \/>\n\\end{eqnarray}<br \/>\n<!--more-->\u7279\u306b\uff0c\\(x = 0\\) \u306e\u307e\u308f\u308a\u306e\u30c6\u30a4\u30e9\u30fc\u5c55\u958b\u306e\u3053\u3068\u3092\u300c\u30de\u30af\u30ed\u30fc\u30ea\u30f3\u5c55\u958b\u300d\u3068\u547c\u3076\uff08\u5834\u5408\u3082\u3042\u308b\u304c\uff0c\\(a\\) \u306e\u5024\u306b\u3088\u3063\u3066\u547c\u3073\u65b9\u304c\u5909\u308f\u308b\u306e\u306f\u899a\u3048\u306b\u304f\u3044\u306e\u3067\uff0c\u8a18\u61b6\u306e\u7c21\u7d20\u5316\u306e\u7acb\u5834\u304b\u3089\uff0c\u3053\u306e\u6388\u696d\u3067\u306f\u7279\u306b\u533a\u5225\u305b\u305a\u306b\u30c6\u30a4\u30e9\u30fc\u5c55\u958b\u3068\u547c\u3076\u3053\u3068\u306b\u3059\u308b\u3002\uff09<\/p>\n<hr \/>\n<p id=\"yui_3_17_2_1_1645509206935_1428\" dir=\"ltr\">\u30c6\u30a4\u30e9\u30fc\u5c55\u958b\u3092\u8aac\u660e\u3059\u308b\u305f\u3081\u306b\u306f\uff0c\u4ee5\u4e0b\u306e\u30c6\u30a4\u30e9\u30fc\u306e\u5b9a\u7406\u304c\u6210\u308a\u7acb\u3064\u3053\u3068\u3092\u8a3c\u660e\u3057\u3066\u304a\u304f\u5fc5\u8981\u304c\u308b\u3002<\/p>\n<h3>\u30c6\u30a4\u30e9\u30fc\u306e\u5b9a\u7406<\/h3>\n<p dir=\"ltr\">\\(f(x)\\) \u304c \\(a, b\\) \u3092\u542b\u3080\u533a\u9593\u3067 \\(n\\) \u56de\u5fae\u5206\u53ef\u80fd\u3067\u3042\u308b\u3068\u304d\uff0c<br \/>\n$$f(b) = f(a) + f'(a)\\,(b-a) + \\frac{f^{&#8221;}(a)}{2!}\\,(b-a)^2 + \\cdots + \\frac{f^{({n-1})}(a)}{(n-1)!} (b-a)^{n-1}+ R_n$$<br \/>\n\u305f\u3060\u3057\uff0c$$R_n = \\frac{f^{({n})}(c)}{n!} \\,(b-a)^{n}$$\u3092\u6e80\u305f\u3059 \\(c\\ \\ (a &lt; c &lt; b) \\) \u304c\u5b58\u5728\u3059\u308b\u3002\u3053\u306e \\(R_n\\) \u3092\u5270\u4f59\u9805\u3068\u547c\u3076\u3002<\/p>\n<h4>\u8a3c\u660e<\/h4>\n<p>\u7c21\u5358\u306e\u305f\u3081\uff0c\\(n=2\\) \u306e\u5834\u5408\uff0c\u3064\u307e\u308a<\/p>\n<p>$$f(b) = f(a) + f'(a)\\,(b-a)\u00a0 +\u00a0 \\frac{f^{&#8221;}(c)}{2!} \\,(b-a)^{2}$$\u3092\u6e80\u305f\u3059 \\(c\\ \\ (a &lt; c &lt; b) \\) \u304c\u5b58\u5728\u3059\u308b\u3053\u3068\u3092\u8a3c\u660e\u3057\u3066\u307f\u308b\u3002\uff08\u4e00\u822c\u306e \\(n\\) \u306e\u5834\u5408\u3082\u540c\u69d8\u3002\uff09<\/p>\n<p>\u8a00\u3044\u63db\u3048\u308b\u3068\uff0c<br \/>\n$$ f(b) -f(a) -f'(a)\\,(b-a)\u00a0 -\\frac{K}{2!}\\,(b-a)^2 = 0$$ \u3068\u306a\u308b\u5b9a\u6570 \\(K\\) \u304c \\(a &lt; c &lt; b\\) \u3067\u3042\u308b \\(c\\) \u3092\u4f7f\u3063\u3066 \\(K = f^{&#8221;}(c)\\) \u3068\u306a\u308b\u3053\u3068\u3092\u793a\u305b\u3070\u3088\u3044\u3002<\/p>\n<p>\u95a2\u6570<br \/>\n$$F(x) \\equiv f(b) -f(x) -f'(x)\\,(b-x)\u00a0 -\\frac{K}{2!}\\,(b-x)^2$$\u3092\u5b9a\u7fa9\u3059\u308b\u3068\uff0c\u5b9a\u7fa9\u304b\u3089\u305f\u3060\u3061\u306b \\(F(b) = 0\\)\u3002<\/p>\n<p>\u307e\u305f\uff0c$K$ \u306e\u5024\u3092\u3046\u307e\u304f\u3068\u308b\u3068\uff0c\u5fc5\u305a $F(a) = 0$ \u3068\u3059\u308b\u3053\u3068\u3082\u3067\u304d\u308b\u306f\u305a\u3067\u3042\u308b\u3002\u8a00\u3044\u63db\u3048\u308b\u3068\uff0c$F(a) = 0$ \u3068\u306a\u308b\u3088\u3046\u306a $K$ \u304c\u5b58\u5728\u3059\u308b\u306f\u305a\u3067\u3042\u308b\u3002<\/p>\n<p>\u305d\u3046\u306a\u308b\u3068 \\(F(a) = F(b)\\) \u3067\u3042\u308b\u3053\u3068\u304b\u3089\uff0c\u95a2\u6570 \\(F(x)\\) \u306f\u533a\u9593 \\((a, b)\\) \u5185\u306e\u3069\u3053\u304b \\(x=c\\) \u3067 \\(F'(c) = 0\\) \u3068\u306a\u308b\u6975\u5024\u3092\u3082\u3064\u306f\u305a\u3067\u3042\u308b\u3002\u3057\u305f\u304c\u3063\u3066\uff0c<\/p>\n<p>\\begin{eqnarray}<br \/>\nF'(x) &amp;=&amp; -f'(x) &#8211; f^{&#8221;}(x)\\, (b-x) + f'(x) + K\\, (b-x) \\\\<br \/>\nF'(c) &amp;=&amp; -f'(c) -f^{&#8221;}(c)\\,(b-c) + f'(c) + K \\,(b-c) = 0\\\\<br \/>\n\\therefore\\ \\ K &amp;=&amp; f^{&#8221;}(c)<br \/>\n\\end{eqnarray}<\/p>\n<p>\u3068\u306a\u308a\uff0c\u8a3c\u660e\u3067\u304d\u305f&#8230;\u3068\u601d\u3046\u3051\u3069\uff0c\u3053\u308c\u3067\u3044\u3044\u304b\u306a\uff1f<\/p>\n<h3>\u30c6\u30a4\u30e9\u30fc\u5c55\u958b<\/h3>\n<p dir=\"ltr\">\u30c6\u30a4\u30e9\u30fc\u306e\u5b9a\u7406\u306b\u304a\u3044\u3066\uff0c\u5270\u4f59\u9805\u304c \\( \\displaystyle \\lim_{n \\rightarrow \\infty} R_n = 0\\) \u306e\u3068\u304d\uff0c\\(f(b)\\) \u306f\u4ee5\u4e0b\u306e\u3088\u3046\u306a\u7121\u9650\u7d1a\u6570\u3067\u3042\u3089\u308f\u3055\u308c\u308b\u3002<br \/>\n$$f(b) = f(a) + f'(a)\\,(b-a) + \\frac{f^{&#8221;}(a)}{2!}\\,(b-a)^2 + \\cdots + \\frac{f^{({n})}(a)}{n!} \\,(b-a)^n+ \\cdots$$<br \/>\n\u3053\u3053\u3067 \\(b = a + x\\) \u3068\u304a\u304f\u3068\uff0c\u4ee5\u4e0b\u306e\u3088\u3046\u306a\uff0c\\(x = a\\) \u306e\u307e\u308f\u308a\u306e\u30c6\u30a4\u30e9\u30fc\u5c55\u958b\u304c\u5f97\u3089\u308c\u308b\u3002<\/p>\n<p dir=\"ltr\">$$f(a+x) = f(a) + f'(a)\\,x + \\frac{f^{&#8221;}(a)}{2!}\\,x^2 + \\cdots + \\frac{f^{({n})}(a)}{n!} \\,x^n+ \\cdots$$<\/p>\n<p dir=\"ltr\">\u79c1\u81ea\u8eab\u306f\uff0c\u4e0a\u8a18\u306e\u3088\u3046\u306b \\(a\\) \u304b\u3089\u5c11\u3057\u96e2\u308c\u305f \\(a + x\\) \u306b\u304a\u3051\u308b\u5024 \\(f(a+x)\\) \u3092 \\(a\\) \u306e\u307e\u308f\u308a\u306e\u30c6\u30a4\u30e9\u30fc\u5c55\u958b\u3067\u8868\u3059\u3068\u3044\u3046\u66f8\u304d\u65b9\u3092\u597d\u3080\u304c\uff0c\u30c6\u30ad\u30b9\u30c8\u306b\u3088\u3063\u3066\u306f\uff0c\\(b = x\\) \u3068\u304a\u3044\u3066\u4ee5\u4e0b\u306e\u8868\u8a18\u3092\u30c6\u30a4\u30e9\u30fc\u5c55\u958b\u3068\u3059\u308b\u3082\u306e\u3082\u591a\u3044\u306e\u3067\uff0c\u4f75\u8a18\u3057\u3066\u304a\u304f\u3002<\/p>\n<p dir=\"ltr\">$$f(x) = f(a) + f'(a)\\,(x-a) + \\frac{f^{&#8221;}(a)}{2!}\\,(x-a)^2 + \\cdots + \\frac{f^{({n})}(a)}{n!} \\,(x-a)^n+ \\cdots$$<\/p>\n<h4>\u30de\u30af\u30ed\u30fc\u30ea\u30f3\u5c55\u958b<\/h4>\n<p dir=\"ltr\">\u7279\u306b\uff0c\\(a = 0\\) \u3064\u307e\u308a \\(x = 0\\) \u306e\u307e\u308f\u308a\u306e\u30c6\u30a4\u30e9\u30fc\u5c55\u958b\u306f\uff0c\u3057\u3070\u3057\u3070\u30de\u30af\u30ed\u30fc\u30ea\u30f3\u5c55\u958b\u3068\u547c\u3070\u308c\uff0c\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u306a\u308b\u3002<\/p>\n<p dir=\"ltr\">$$f(x) = f(0) + f'(0)\\,x + \\frac{f^{&#8221;}(0)}{2!}\\,x^2 + \\cdots + \\frac{f^{({n})}(0)}{n!} \\,x^n+ \\cdots$$<\/p>\n<h3>\u30c6\u30a4\u30e9\u30fc\u5c55\u958b\u306e\u4f8b<\/h3>\n<h4>$f(x) = (1 + x)^p$ \u306e\u30c6\u30a4\u30e9\u30fc\u5c55\u958b<\/h4>\n<p>\\(f(x) = (1+x)^p\\) \u3068\u3044\u3046\u3079\u304d\u95a2\u6570\uff08\u300c\u3068\u8a00\u3046\u3079\u304d\u300d\u95a2\u6570\uff0c\u3067\u306f\u306a\u304f\u3066\u300c\u3068\u3044\u3046\u300d\u51aa\uff08\u3079\u304d\uff09\u95a2\u6570\uff09\u306e\u5834\u5408\uff0c\\(f'(x) = p (1+x)^{p-1}\\) \u3067\u3042\u308b\u304b\u3089 \\(x = 0\\) \u306e\u307e\u308f\u308a\u306e\u30c6\u30a4\u30e9\u30fc\u5c55\u958b\u306f<\/p>\n<p>\\begin{eqnarray}<br \/>\nf(x) &amp;=&amp; f(0) + f'(0)\\, x + \\cdots \\\\<br \/>\n&amp;=&amp; 1 + p\\, x\u00a0 + \\cdots<br \/>\n\\end{eqnarray}<\/p>\n<p>\u305f\u3068\u3048\u3070 $p=2$ \u306e\u5834\u5408\u306f<\/p>\n<p>$$(1 + x)^2 = 1 + 2 x + \\cdots$$<\/p>\n<p>\u3068\u306a\u308b\u304c\uff0c\u3053\u308c\u306f\u5c55\u958b\u5f0f $(1 + x)^2 = 1 + 2 x + x^2$ \u306e $x^2$ \u306e\u9805\u3092\u7701\u7565\u3057\u305f\u5f62\u306b\u306a\u3063\u3066\u3044\u308b\u3057\uff0c$p= -1$ \u306e\u5834\u5408\u306f<\/p>\n<p>$$(1 + x)^{-1} = \\frac{1}{1 + x} = 1 -x + \\cdots$$ \u3068\u306a\u308b\u3002<\/p>\n<p>\u3053\u308c\u306f\u7c21\u5358\u306a\u516c\u5f0f\u3068\u3057\u3066\uff0c\u810a\u9ac4\u53cd\u5c04\u3067\u51fa\u3066\u304f\u308b\u3088\u3046\u306b\u899a\u3048\u3066\u304a\u3044\u3066\u304f\u3060\u3055\u3044\u3002\u3053\u3093\u306a\u306e\u793e\u4f1a\u306b\u51fa\u305f\u3089\u4f7f\u308f\u306a\u3044\u3063\u3066\uff1f\u305d\u3093\u306a\u3053\u3068\u306a\u3044\u3067\u3059\u3088\uff01\uff08\u305f\u3076\u3093\u3002\uff09<\/p>\n<p>\u4eca\u5f8c\u306e\u6388\u696d\u3067\u3082\uff0c\u3053\u306e\u30c6\u30a4\u30e9\u30fc\u5c55\u958b\u306f\u3044\u308d\u3093\u306a\u3068\u3053\u308d\u3067\u4f7f\u3044\u307e\u3059\u3002\u4f8b\u3048\u3070\uff0c\u96fb\u78c1\u6c17\u5b66\u306a\u3089<\/p>\n<ul>\n<li><a href=\"https:\/\/home.hirosaki-u.ac.jp\/relativity\/%e9%9b%bb%e7%a3%81%e6%b0%97%e5%ad%a6-i\/%e9%9d%99%e9%9b%bb%e5%a0%b4%ef%bc%9a%e9%9b%bb%e8%8d%b7%e5%af%86%e5%ba%a6%e3%81%8b%e3%82%89%e7%9b%b4%e6%8e%a5%e9%9b%bb%e5%a0%b4%e3%82%92%e6%b1%82%e3%82%81%e3%82%8b\/#i-4\" target=\"_blank\" rel=\"noopener\">\u96fb\u6c17\u53cc\u6975\u5b50\u306b\u3088\u308b\u96fb\u5834<\/a><\/li>\n<li><a href=\"https:\/\/home.hirosaki-u.ac.jp\/relativity\/%e9%9b%bb%e7%a3%81%e6%b0%97%e5%ad%a6-i\/%e9%9d%99%e7%a3%81%e5%a0%b4%ef%bc%9a%e9%9b%bb%e6%b5%81%e5%af%86%e5%ba%a6%e3%81%8b%e3%82%89%e7%9b%b4%e6%8e%a5%e9%9d%99%e7%a3%81%e5%a0%b4%e3%82%92%e6%b1%82%e3%82%81%e3%82%8b\/#i-4\" target=\"_blank\" rel=\"noopener\">\u5fae\u5c0f\u306a\u5186\u96fb\u6d41\u306b\u3088\u308b\u78c1\u5834\uff1a\u78c1\u6c17\u53cc\u6975\u5b50<\/a><\/li>\n<\/ul>\n<p>\u306a\u3069\uff0c\u3044\u308d\u3093\u306a\u3068\u3053\u308d\u306b\u51fa\u3066\u304d\u307e\u3059\u3002\u4ee5\u4e0b\u306e\u540d\u8a00\uff08\u8ff7\u8a00\uff09\u3068\u3057\u3066\u899a\u3048\u3066\u304a\u3044\u305f\u3089\u3069\u3046\u3067\u3057\u3087\u3046\u304b\u3002<\/p>\n<p style=\"text-align: center;\"><span style=\"font-family: helvetica, arial, sans-serif;\"><strong>\u4eba\u751f\u306e\u516b\u5272\u304c\u305f\u306f\uff0c\u30c6\u30a4\u30e9\u30fc\u5c55\u958b\u3067\u7406\u89e3\u3067\u304d\u308b\u3002<\/strong><\/span><\/p>\n<h4>\u6307\u6570\u95a2\u6570\u306e\u30c6\u30a4\u30e9\u30fc\u5c55\u958b<\/h4>\n<p>\\(f(x) = e^x\\) \u306e\u5834\u5408\uff0c\\(f^{(n)}(x) = e^x\\)\u00a0 \u3067\u3042\u308b\u304b\u3089 \\(x = 0\\) \u306e\u307e\u308f\u308a\u306e\u30c6\u30a4\u30e9\u30fc\u5c55\u958b\u306f<\/p>\n<p>\\begin{eqnarray}<br \/>\ne^x &amp;=&amp; f(0) + f'(0)\\, x + \\frac{f^{&#8221;}(0)}{2!}\\,x^2 + \\cdots \\\\<br \/>\n&amp;=&amp; 1 + x + \\frac{x^2}{2!} + \\frac{x^3}{3!} + \\cdots \\\\<br \/>\n&amp;=&amp; \\sum_{n=0}^{\\infty} \\frac{x^n}{n!}<br \/>\n\\end{eqnarray}<\/p>\n<p>\u3061\u306a\u307f\u306b\uff0c\u3053\u306e\u5f0f\u3067 \\(x = 1\\) \u3068\u304a\u304f\u3068<br \/>\n$$ e = \\sum_{n=0}^{\\infty} \\frac{1}{n!}$$ \u3068\u306a\u308a\uff0c\u3053\u306e\u7d1a\u6570\u5c55\u958b\u3092\u4f7f\u3063\u3066\u30cd\u30a4\u30d4\u30a2\u6570\u306e\u8fd1\u4f3c\u5024\u3092\u6c42\u3081\u308b\u3053\u3068\u304c\u3067\u304d\u308b\u3002<br \/>\n\\begin{eqnarray}<br \/>\n\\sum_{n=0}^{3} \\frac{1}{n!}\u00a0 = 1+1+\\frac{1}{2} + \\frac{1}{6}&amp;\\simeq&amp; 2.66667 \\\\<br \/>\n\\sum_{n=0}^{5} \\frac{1}{n!}\u00a0 = 1 + 1+ \\frac{1}{2} + \\cdots + \\frac{1}{120}&amp;\\simeq&amp; 2.71667 \\\\<br \/>\n\\sum_{n=0}^{10} \\frac{1}{n!}\u00a0 = 1 + 1+ \\frac{1}{2} + \\cdots + \\frac{1}{3628800}&amp;\\simeq&amp;2.71828<br \/>\n\\end{eqnarray}<\/p>\n<h4>\u4e09\u89d2\u95a2\u6570\u306e\u30c6\u30a4\u30e9\u30fc\u5c55\u958b<\/h4>\n<p>\\(f(x) = \\sin x\\) \u306e\u5834\u5408\u306e \\(x = 0\\) \u306e\u307e\u308f\u308a\u306e\u30c6\u30a4\u30e9\u30fc\u5c55\u958b\u306f<br \/>\n\\begin{eqnarray}<br \/>\n\\sin x &amp;=&amp; f(0) + f'(0)\\, x + \\frac{f^{&#8221;}(0)}{2!}\\,x^2 + \\frac{f^{&#8221;&#8217;}(0)}{3!}\\,x^3 +\\cdots \\\\<br \/>\n&amp;=&amp; \\sin 0 + \\cos 0 \\cdot x + \\frac{-\\sin 0}{2!}\\,x^2 + \\frac{-\\cos 0}{3!}\\,x^3 + \\cdots \\\\<br \/>\n&amp;=&amp; x -\\frac{x^3}{3!} + \\frac{x^5}{5!} -\\frac{x^7}{7!} + \\cdots\\\\<br \/>\n&amp;=&amp; \\sum_{n=0}^{\\infty} \\frac{(-1)^n}{(2n+1)!}\\,x^{2n+1}<br \/>\n\\end{eqnarray}<\/p>\n<p>\\(f(x) = \\cos x\\) \u306e\u5834\u5408\u306e \\(x = 0\\) \u306e\u307e\u308f\u308a\u306e\u30c6\u30a4\u30e9\u30fc\u5c55\u958b\u306f<br \/>\n\\begin{eqnarray}<br \/>\n\\cos x &amp;=&amp; f(0) + f'(0)\\, x + \\frac{f^{&#8221;}(0)}{2!}\\,x^2 + \\frac{f^{&#8221;&#8217;}(0)}{3!}\\,x^3 +\\cdots \\\\<br \/>\n&amp;=&amp; \\cos 0 -\\sin 0 \\cdot x + \\frac{-\\cos 0}{2!}\\,x^2 + \\frac{\\sin 0}{3!}\\,x^3 + \\cdots \\\\<br \/>\n&amp;=&amp; 1 -\\frac{x^2}{2!} + \\frac{x^4}{4!} -\\frac{x^6}{6!} + \\cdots\\\\<br \/>\n&amp;=&amp; \\sum_{n=0}^{\\infty} \\frac{(-1)^n}{(2n)!}\\,x^{2n}<br \/>\n\\end{eqnarray}<\/p>\n<p>\\(\\sin x\\) \u306f\u5947\u95a2\u6570\u3067\u3042\u308b\u3053\u3068\u304b\u3089\uff0c\\(x\\) \u306e\u5947\u6570\u4e57\u306e\u3079\u304d\u95a2\u6570\u306e\u307f\u306e\u7d1a\u6570\u5c55\u958b\u3068\u3057\u3066\u3042\u3089\u308f\u3055\u308c\uff0c\\(\\cos x\\) \u306f\u5076\u95a2\u6570\u3067\u3042\u308b\u3053\u3068\u304b\u3089\uff0c\\(x\\) \u306e\u5076\u6570\u4e57\u306e\u3079\u304d\u95a2\u6570\u306e\u307f\u306e\u7d1a\u6570\u5c55\u958b\u3068\u3057\u3066\u3042\u3089\u308f\u3055\u308c\u3066\u3044\u308b\u3053\u3068\u304c\u308f\u304b\u308b\u3002<\/p>\n<h4>\u4e09\u89d2\u95a2\u6570\u306e\u6975\u9650\u516c\u5f0f\u306e\u78ba\u8a8d<\/h4>\n<p>\u4e09\u89d2\u95a2\u6570\u306e\u5fae\u5206\u3092\u5c0e\u304f\u969b\u306b\u4ee5\u4e0b\u306e\u6975\u9650\u516c\u5f0f\u3092\u7528\u3044\u305f\u3002<\/p>\n<p>$$\\lim_{x \\rightarrow 0} \\frac{\\sin x}{x} = 1, \\quad \\lim_{x \\rightarrow 0} \\frac{1-\\cos x}{x} = 0 $$<\/p>\n<p>\u3053\u308c\u3089\u3082\uff0c\\(x=0\\) \u306e\u307e\u308f\u308a\u306e\u30c6\u30a4\u30e9\u30fc\u5c55\u958b\u3092\u4f7f\u3046\u3068\uff0c\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u3057\u3066\u78ba\u8a8d\u3067\u304d\u308b\u3002\uff08\u3042\u304f\u307e\u3067\u78ba\u8a8d\u3067\u3042\u3063\u3066\u8a3c\u660e\u3067\u306f\u306a\u3044\u3002\u306a\u305c\u306a\u3089\uff0c\u30c6\u30a4\u30e9\u30fc\u5c55\u958b\u3067\u306f\u4e09\u89d2\u95a2\u6570\u306e\u5fae\u5206\u3092\u4f7f\u3046\u304c\uff0c\u305d\u306e\u5fae\u5206\u306f\u4e0a\u8a18\u306e\u6975\u9650\u516c\u5f0f\u3092\u7528\u3044\u3066\u5c0e\u304b\u308c\u3066\u3044\u308b\u304b\u3089\u3002\uff09<\/p>\n<p>\\begin{eqnarray}<br \/>\n\\lim_{x \\rightarrow 0} \\frac{\\sin x}{x} &amp;=&amp; \\lim_{x \\rightarrow 0}\\frac{ x -{\\displaystyle\\frac{x^3}{3!}} + O(x^5)}{x} \\\\<br \/>\n&amp;=&amp; \\lim_{x \\rightarrow 0}\\left\\{ 1 -\\frac{x^2}{3!} + O(x^4)\\right\\}\\\\<br \/>\n&amp;=&amp; 1<br \/>\n\\end{eqnarray}<\/p>\n<p>\\begin{eqnarray}<br \/>\n\\lim_{x \\rightarrow 0} \\frac{1-\\cos x}{x} &amp;=&amp; \\lim_{x \\rightarrow 0}\\frac{ 1 -\\left(1 -{\\displaystyle \\frac{x^2}{2!}} + O(x^4)\\right)}{x} \\\\<br \/>\n&amp;=&amp; \\lim_{x \\rightarrow 0}\\left\\{ \\frac{x^2}{2!} + O(x^3)\\right\\}\\\\<br \/>\n&amp;=&amp; 0<br \/>\n\\end{eqnarray}<\/p>\n<p>\u3053\u3053\u3067\uff0c\u4f8b\u3048\u3070\u5927\u6587\u5b57\u306e\u30aa\u30fc\u3092\u4f7f\u3063\u305f \\(O(x^5)\\) \u306e\u8868\u8a18\u306f\uff0c\\(x^5\\) \u4ee5\u4e0a\u306e\u9ad8\u6b21\u306e\u9805\u3068\u3044\u3046\u610f\u5473\u3092\u8868\u3059\u3002\uff08\u306a\u304a\uff0c\u30c6\u30ad\u30b9\u30c8\u306b\u3088\u3063\u3066\u306f\u5c0f\u6587\u5b57\u306e\u30aa\u30fc\u3092\u4f7f\u3063\u3066 \\(o(x^4)\\) \u306a\u3069\u3068\u66f8\u304f\u5834\u5408\u3082\u591a\u3044\u304c\uff0c\u3053\u308c\u306f \\(x^4\\) \u3088\u308a\u3082\u9ad8\u6b21\u306e\u9805\u3068\u610f\u5473\u306b\u306a\u308b\u3089\u3057\u3044\u3002\uff09<\/p>\n<h4>\u30ed\u30d4\u30bf\u30eb\u306e\u5b9a\u7406\u306e\u78ba\u8a8d<\/h4>\n<p>\\(f(a) = 0, g(a) = 0\\) \u306e\u3068\u304d\uff0c\u6975\u9650\u5024 \\(\\displaystyle \\lim_{x \\rightarrow a}\\frac{f(x)}{g(x)}\\) \u306f \\(\\displaystyle \\frac{0}{0}\\) \u578b\u306e\u4e0d\u5b9a\u5f62\u306b\u898b\u3048\u308b\u304c\uff0c\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u3057\u3066\u6975\u9650\u3092\u6c42\u3081\u308b\u3053\u3068\u304c\u3067\u304d\u308b\u3068\u3044\u3046\u306e\u304c\u30ed\u30d4\u30bf\u30eb\u306e\u5b9a\u7406\u3067\u3042\u3063\u305f\u3002<\/p>\n<p>$$\\lim_{x \\rightarrow a}\\frac{f(x)}{g(x)} = \\lim_{x \\rightarrow a}\\frac{f'(x)}{g'(x)}$$<\/p>\n<p>\u3053\u308c\u3082\u30c6\u30a4\u30e9\u30fc\u5c55\u958b\u3092\u4f7f\u3063\u3066\uff0c\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u78ba\u8a8d\u3059\u308b\u3053\u3068\u304c\u3067\u304d\u308b\u3002\uff08\u8a3c\u660e\u3068\u3044\u3046\u308f\u3051\u3067\u306f\u306a\u3044\uff0c\u5ff5\u306e\u305f\u3081\u3002\uff09<\/p>\n<p id=\"yui_3_17_2_1_1645509206935_1434\">\\begin{eqnarray}<br \/>\n\\lim_{x \\rightarrow a}\\frac{f(x)}{g(x)} &amp;=&amp; \\lim_{x \\rightarrow a}\\frac{f(a)+ f'(a) \\,(x-a) + \\cdots}{g(a)+ g'(a) \\,(x-a) + \\cdots}\\\\<br \/>\n&amp;=&amp; \\lim_{x \\rightarrow a}\\frac{ f'(a) \\,(x-a) + \\cdots}{g'(a) \\,(x-a) + \\cdots}\\\\<br \/>\n&amp;=&amp; \\lim_{x \\rightarrow a}\\frac{ f'(a) }{g'(a) }\\\\<br \/>\n&amp;=&amp; \\lim_{x \\rightarrow a}\\frac{ f'(x) }{g'(x) }<br \/>\n\\end{eqnarray}<\/p>\n","protected":false},"excerpt":{"rendered":"<p>\\(f(x)\\) \u304c \\(x = a\\) \u3092\u542b\u3080\u533a\u9593\u3067\u5fae\u5206\u53ef\u80fd\u3067\u3042\u308b\u3068\u304d\uff0c\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u5c55\u958b\u3067\u304d\u308b\u3002\u3053\u308c\u3092\u300c\\(x = a\\) \u306e\u307e\u308f\u308a\u306e\u30c6\u30a4\u30e9\u30fc\u5c55\u958b\u300d\u3068\u547c\u3076\u3002 \\begin{eqnarray} f(a + x) &amp;=&amp; f(a) + f'(a)\\, x + \\frac{f^{&#8221;}(a)}{2!}\\, x^2 + \\cdots + \\frac{f^{({n})}(a)}{n!}\\, x^n + \\cdots\\\\ &amp;=&amp; f(a) + \\sum_{k=1} \\frac{f^{(k)}(a)}{k!}\\, x^k \\end{eqnarray}<\/p><p><a class=\"more-link btn\" href=\"https:\/\/home.hirosaki-u.ac.jp\/relativity\/%e7%90%86%e5%b7%a5%e7%b3%bb%e3%81%ae%e6%95%b0%e5%ad%a6b\/%e3%83%86%e3%82%a4%e3%83%a9%e3%83%bc%e5%b1%95%e9%96%8b\/\">\u7d9a\u304d\u3092\u8aad\u3080<\/a><\/p>\n","protected":false},"author":33,"featured_media":0,"parent":2068,"menu_order":12,"comment_status":"closed","ping_status":"closed","template":"","meta":{"inline_featured_image":false,"footnotes":""},"class_list":["post-2139","page","type-page","status-publish","hentry","nodate","item-wrap"],"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/2139","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/users\/33"}],"replies":[{"embeddable":true,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/comments?post=2139"}],"version-history":[{"count":25,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/2139\/revisions"}],"predecessor-version":[{"id":8725,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/2139\/revisions\/8725"}],"up":[{"embeddable":true,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/2068"}],"wp:attachment":[{"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/media?parent=2139"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}