{"id":6425,"date":"2023-06-01T15:22:11","date_gmt":"2023-06-01T06:22:11","guid":{"rendered":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/?page_id=6425"},"modified":"2024-05-04T13:14:26","modified_gmt":"2024-05-04T04:14:26","slug":"%e5%8f%82%e8%80%83%ef%bc%9a%e3%83%ad%e3%83%94%e3%82%bf%e3%83%ab%e3%81%ae%e5%ae%9a%e7%90%86","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\/%e5%ba%83%e7%be%a9%e3%81%ae%e7%a9%8d%e5%88%86\/%e5%8f%82%e8%80%83%ef%bc%9a%e3%83%ad%e3%83%94%e3%82%bf%e3%83%ab%e3%81%ae%e5%ae%9a%e7%90%86\/","title":{"rendered":"\u53c2\u8003\uff1a\u30ed\u30d4\u30bf\u30eb\u306e\u5b9a\u7406"},"content":{"rendered":"

\\(\\displaystyle \\lim_{x \\rightarrow a}\\frac{f(x)}{g(x)} \\) \u304c \\(\\displaystyle \\frac{0}{0}\\) \u3084 \\(\\displaystyle \\frac{\\infty}{\\infty}\\) \u306e\u3088\u3046\u306a\u4e0d\u5b9a\u5f62\u306b\u306a\u308b\u3068\u304d\u306b \\(\\displaystyle \\lim_{x \\rightarrow a} \\frac{f(x)}{g(x)} \\) \u3092\u6c42\u3081\u308b\u65b9\u6cd5\u3002<\/p>\n

\u8a3c\u660e\u306a\u3057\u306b\u66f8\u3044\u3066\u304a\u304f\u3068\uff0c\\(f(a) = 0, \\ g(a) = 0\\) \u306e\u3068\u304d\uff0c<\/p>\n

$$\\lim_{x \\rightarrow a} \\frac{f(x)}{g(x)} = \\frac{f'(a)}{g'(a)}$$<\/p>\n

\u307e\u305f\uff0c\\(f(a) \\rightarrow \\pm \\infty, \\ g(a) = \\rightarrow \\pm \\infty\\) \u306e\u3068\u304d\u3082\u540c\u69d8\u306b\uff0c<\/p>\n

$$\\lim_{x \\rightarrow a} \\frac{f(x)}{g(x)} = \\frac{f'(a)}{g'(a)}$$<\/p>\n

\u96d1\u8ac7\uff1a<\/h4>\n

\u3061\u306a\u307f\u306b\uff0c\\(\\displaystyle \\lim_{x \\rightarrow 0} \\frac{\\sin x}{x}=1\\) \u3082\u3053\u306e\u30ed\u30d4\u30bf\u30eb\u306e\u5b9a\u7406\u3092\u4f7f\u3063\u3066<\/p>\n

$$\\lim_{x \\rightarrow 0} \\frac{\\sin x}{x} = \\frac{\\cos 0}{1} = 1$$<\/p>\n

\u306e\u3088\u3046\u306b\u8a3c\u660e\u3067\u304d\u308b\uff0c\u3068\u3044\u3046\u306e\u306f\u30c0\u30e1\u3002\u306a\u305c\u306a\u3089\uff0c\u4e0a\u8a18\u306e\u8a08\u7b97\u306e\u969b\u306b<\/p>\n

$$\\left( \\sin x \\right)’ = \\cos x$$<\/p>\n

\u3092\u4f7f\u3063\u3066\u3044\u308b\u304c\uff0c\u3053\u306e\u4e09\u89d2\u95a2\u6570\u306e\u5fae\u5206\u3092\u793a\u3059\u969b\u306b\uff0c\\(\\displaystyle \\lim_{x \\rightarrow 0} \\frac{\\sin x}{x}=1\\) \u304c\u4f7f\u308f\u308c\u3066\u3044\u308b\u304b\u3089\u3002\u300c\u4e09\u89d2\u95a2\u6570\u306e\u5fae\u5206<\/a><\/strong><\/span>\u300d\u306e\u30da\u30fc\u30b8\u3092\u53c2\u7167\u3002<\/p>\n

\u304f\u3069\u3044\u3088\u3046\u3060\u304c\uff0c\\(\\displaystyle \\lim_{x \\rightarrow 0} \\frac{\\sin x}{x}=1\\) \u3092\u793a\u3059\u305f\u3081\u306b\u30ed\u30d4\u30bf\u30eb\u306e\u5b9a\u7406\u3092\u4f7f\u304a\u3046\u3068\u3059\u308b\u3068\uff0c\u305d\u306e\u969b\u306b \\(\\left( \\sin x \\right)’ = \\cos x\\) \u304c\u5fc5\u8981\u306b\u306a\u308b\u304c\uff0c\\(\\left( \\sin x \\right)’ = \\cos x\\)\u00a0 \u3092\u793a\u3059\u305f\u3081\u306b\u306f \\(\\displaystyle \\lim_{x \\rightarrow 0} \\frac{\\sin x}{x}=1\\) \u3092\u4f7f\u3046\u5fc5\u8981\u304c\u3042\u308b\u306e\u3067\u5802\u3005\u5de1\u308a\u306b\u306a\u3063\u3066\uff0c\u8a3c\u660e\u306b\u306a\u3089\u306a\u3044\u3068\u3044\u3046\u3053\u3068\u3002<\/p>\n

\\(\\displaystyle \\lim_{x \\rightarrow 0} \\frac{\\sin x}{x}=1\\) \u306f\u4e09\u89d2\u95a2\u6570\u306e\u5fae\u5206\u3092\u4f7f\u308f\u305a\u306b\u8a3c\u660e\u3059\u308b\u5fc5\u8981\u304c\u3042\u308b\u3002\u3068\u3044\u3046\u3053\u3068\u3067\uff0c\u300c<\/span>\u53c2\u8003\uff1a\u4e09\u89d2\u95a2\u6570\u306e\u6975\u9650\u516c\u5f0f\u306e\u8a3c\u660e<\/span><\/a><\/strong>\u300d\u306b\u305d\u306e\u8a3c\u660e\u3092\u66f8\u3044\u3066\u304a\u3044\u305f\u306e\u3067\u3042\u3063\u305f\u3002<\/p>\n

\u30ed\u30d4\u30bf\u30eb\u306e\u5b9a\u7406\u3092\u30c6\u30a4\u30e9\u30fc\u5c55\u958b\u7684\u306b\u7406\u89e3\u3059\u308b<\/h4>\n

\\(f(a) = 0, \\ g(a) = 0\\) \u306e\u3068\u304d\uff0c<\/p>\n

$$\\lim_{x \\rightarrow a} \\frac{f(x)}{g(x)} = \\frac{f'(a)}{g'(a)}$$<\/p>\n

\u3067\u3042\u308b\u3053\u3068\u3092\u30c6\u30a4\u30e9\u30fc\u5c55\u958b\u3092\u4f7f\u3063\u3066\uff08\u8a3c\u660e\u3067\u306f\u306a\u304f\uff09\u7406\u89e3\u3057\u3066\u307f\u308b\u3002<\/p>\n

\\begin{eqnarray}
\n\\lim_{x \\rightarrow a} \\frac{f(x)}{g(x)} &=& \\lim_{x \\rightarrow 0} \\frac{f(a+x)}{g(a+x)}\\\\
\n&=& \\lim_{x \\rightarrow 0} \\frac{f(a) + f'(a) x + \\cdots}{g(a) + g'(a) x + \\cdots}\\\\
\n&=& \\lim_{x \\rightarrow 0} \\frac{0 + f'(a) x + O(x^2)}{0 + g'(a) x + O(x^2)}\\\\
\n&=& \\lim_{x \\rightarrow 0} \\frac{f'(a)\u00a0 + O(x)}{g'(a)\u00a0 + O(x)}\\\\
\n&=& \\frac{f'(a) }{g'(a) }
\n\\end{eqnarray}<\/p>\n

\u305f\u3060\u3057\uff0c\u3053\u306e\u307e\u307e\u3067\u306f \\(f(a) \\rightarrow \\pm \\infty, \\ g(a) = \\rightarrow \\pm \\infty\\) \u306e\u3068\u304d\u3082\uff0c<\/p>\n

$$\\lim_{x \\rightarrow a} \\frac{f(x)}{g(x)} = \\frac{f'(a)}{g'(a)}$$<\/p>\n

\u3068\u306a\u308b\u3053\u3068\u304c\u3046\u307e\u304f\u7406\u89e3\u3067\u304d\u306a\u304f\u3066\u60a9\u3080\u306a\u3041\u3002<\/p>\n

\u8ffd\u8a18\uff1a<\/p>\n

\u60a9\u3093\u3067\u3070\u304b\u308a\u3082\u3044\u3089\u308c\u306a\u3044\u306e\u3067\uff0c\u306a\u3093\u3068\u304b \\(f(a) \\rightarrow \\pm \\infty, \\ g(a) = \\rightarrow \\pm \\infty\\) \u306e\u3068\u304d\u3082<\/p>\n

$$\\lim_{x \\rightarrow a} \\frac{f(x)}{g(x)} = \\frac{f'(a)}{g'(a)}$$<\/p>\n

\u3068\u306a\u308b\u3053\u3068\u3092\u30c6\u30a4\u30e9\u30fc\u5c55\u958b\u7684\u306b\u7406\u89e3\u3057\u3066\u307f\u308b\u3002<\/p>\n

$$F(x) \\equiv \\frac{1}{f(x)}, \\quad G(x) \\equiv \\frac{1}{g(x)}$$<\/p>\n

\u3068\u5b9a\u7fa9\u3059\u308b\u3068\uff0c$F(a) = 0, \\quad G(a) = 0$ \u3067\u3042\u308b\u304b\u3089<\/p>\n

\\begin{eqnarray}
\n\\lim_{x \\rightarrow a} \\frac{f(x)}{g(x)} = \\lim_{x \\rightarrow 0} \\frac{f(a+x)}{g(a+x)}
\n&=& \\lim_{x \\rightarrow 0} \\frac{G(a+x)}{F(a+x)} \\\\
\n&=& \\lim_{x \\rightarrow 0} \\frac{G(a) + G'(a) x + O(x^2)}{F(a) + F'(a) x + O(x^2)}\\\\
\n&=& \\lim_{x \\rightarrow 0} \\frac{G'(a) + O(x)}{F'(a)\u00a0 + O(x)}\\\\
\n&=& \\lim_{x \\rightarrow 0} \\frac{G'(a+x)}{F'(a+x) }\\\\
\n&=& \\lim_{x \\rightarrow 0} \\frac{-\\frac{g'(a+x)}{(g(a+x))^2}}{-\\frac{f'(a+x)}{(f(a+x))^2}}\\\\
\n&=& \\left(\\lim_{x \\rightarrow 0} \\frac{f(a+x)}{g(a+x)}\u00a0 \\right)^2\\ \\frac{g'(a)}{f'(a)} \\\\
\n\\therefore\\ \\ 1 &=& \\lim_{x \\rightarrow 0} \\frac{f(a+x)}{g(a+x)} \\ \\frac{g'(a)}{f'(a)} \\\\
\n\\therefore\\ \\ \\lim_{x \\rightarrow 0} \\frac{f(a+x)}{g(a+x)} = \\lim_{x \\rightarrow a} \\frac{f(x)}{g(x)} &=& \\frac{f'(a)} {g'(a)}
\n\\end{eqnarray}<\/p>\n

… \u3068\u3044\u3046\u3075\u3046\u306a\u7406\u89e3\uff08\u8a3c\u660e\u3067\u306f\u306a\u304f\uff09\u304c\u53ef\u80fd\u304b\u3068\u3002<\/p>\n","protected":false},"excerpt":{"rendered":"

\\(\\displaystyle \\lim_{x \\rightarrow a}\\frac{f(x)}{g(x)} \\) \u304c \\(\\displaystyle \\frac{0}{0}\\) \u3084 \\(\\displaystyle \\frac{\\infty}{\\infty}\\) \u306e\u3088\u3046\u306a\u4e0d\u5b9a\u5f62\u306b\u306a\u308b\u3068\u304d\u306b \\(\\displaystyle \\lim_{x \\rightarrow a} \\frac{f(x)}{g(x)} \\) \u3092\u6c42\u3081\u308b\u65b9\u6cd5\u3002<\/p>

\u7d9a\u304d\u3092\u8aad\u3080<\/a><\/p>\n","protected":false},"author":33,"featured_media":0,"parent":3274,"menu_order":10,"comment_status":"closed","ping_status":"closed","template":"","meta":{"inline_featured_image":false,"footnotes":""},"class_list":["post-6425","page","type-page","status-publish","hentry","nodate","item-wrap"],"aioseo_notices":[],"aioseo_head":"\n\t\t\n\t\n\t\n\t\n\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t