\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":[],"_links":{"self":[{"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/6425","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=6425"}],"version-history":[{"count":6,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/6425\/revisions"}],"predecessor-version":[{"id":8538,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/6425\/revisions\/8538"}],"up":[{"embeddable":true,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/3274"}],"wp:attachment":[{"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/media?parent=6425"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}