{"id":813,"date":"2013-06-09T21:16:16","date_gmt":"2013-06-09T12:16:16","guid":{"rendered":"http:\/\/www.roundown.net\/nyushi\/?p=813"},"modified":"2021-09-09T07:05:39","modified_gmt":"2021-09-08T22:05:39","slug":"osr201301","status":"publish","type":"post","link":"https:\/\/www.roundown.net\/nyushi\/osr201301\/","title":{"rendered":"\u962a\u5927\u7406\u7cfb2013\uff1a\u7b2c1\u554f"},"content":{"rendered":"<hr \/>\n<p>\u4e09\u89d2\u95a2\u6570\u306e\u6975\u9650\u306b\u95a2\u3059\u308b\u516c\u5f0f\r\n\\[\r\n\\displaystyle\\lim _ {x \\rightarrow 0} \\dfrac{\\sin x}{x} = 1\r\n\\]\r\n\u3092\u793a\u3059\u3053\u3068\u306b\u3088\u308a, \\(\\sin x\\) \u306e\u5c0e\u95a2\u6570\u304c \\(\\cos x\\) \u3067\u3042\u308b\u3053\u3068\u3092\u8a3c\u660e\u305b\u3088.<\/p>\r\n<hr \/>\r\n<!--more-->\r\n<h2>\u3010 \u89e3 \u7b54 \u3011<\/h2>\r\n<p><img loading=\"lazy\" decoding=\"async\" src=\"\/\/www.roundown.net\/nyushi\/wp-content\/uploads\/osaka_r_2013_01_011.png\" alt=\"osaka_r_2013_01_01\" width=\"215\" height=\"190\" class=\"aligncenter size-full wp-image-819\" \/><\/p>\r\n<p>\u4e0a\u56f3\u306e\u3088\u3046\u306a\u5358\u4f4d\u5186\u3092\u8003\u3048, A \\((1,0)\\) , B \\(( \\cos x , \\sin x )\\) , C \\(( 1 , \\tan x ) \\ \\left( 0 \\lt x \\lt \\dfrac{\\pi}{2} \\right)\\) \u3068\u304a\u304f.<br \/>\r\n\u3053\u306e\u3068\u304d, \\(3\\) \u3064\u306e\u56f3\u5f62 \u25b3OAB , \u25b3OAC , \u6247\u5f62 OAB \u306e\u9762\u7a4d\u3092\u305d\u308c\u305e\u308c \\(S _ 1 , S _ 2 , S\\) \u3068\u304a\u304f\u3068\r\n\\[\r\nS _ 1 \\lt S \\lt S _ 2\r\n\\]\r\n\u304c\u6210\u7acb\u3059\u308b. \u305d\u308c\u305e\u308c\u3092 \\(x\\) \u3092\u7528\u3044\u3066\u8868\u305b\u3070\r\n\\[\\begin{align}\r\n\\dfrac{1}{2} \\cdot 1^2 \\cdot \\sin x & \\lt \\dfrac{1}{2} \\cdot 1^2 \\cdot x \\lt \\dfrac{1}{2} \\cdot 1 \\cdot \\tan x \\\\\r\n\\sin x & \\lt x \\lt \\tan x \\\\\r\n\\text{\u2234} \\quad 1 & \\lt \\dfrac{x}{\\sin x} \\lt \\dfrac{1}{\\cos x} \\quad ( \\ \\text{\u2235} \\ \\sin x \\gt 0 \\ ) \\ .\r\n\\end{align}\\]\r\n\u3053\u3053\u3067, \\(\\displaystyle\\lim _ {x \\rightarrow +0} \\dfrac{1}{\\cos x} = 1\\) \u306a\u306e\u3067, \u306f\u3055\u307f\u3046\u3061\u306e\u539f\u7406\u3088\u308a\r\n\\[\\begin{align}\r\n\\displaystyle\\lim _ {x \\rightarrow +0} \\dfrac{x}{\\sin x} & = 1 \\\\\r\n\\text{\u2234} \\quad \\displaystyle\\lim _ {x \\rightarrow +0} \\dfrac{\\sin x}{x} & = 1 \\quad ... [1] \\ .\r\n\\end{align}\\]\r\n\u3064\u3065\u3044\u3066, \\(x \\lt 0\\) \u306e\u5834\u5408\u306b\u3064\u3044\u3066\u8003\u3048\u308b.<br \/>\r\n\\(x = -t\\) \u3068\u304a\u304f\u3068, \\(t \\gt 0\\) \u3067\u3042\u308a\r\n\\[\r\n\\dfrac{\\sin x}{x} = \\dfrac{- \\sin t}{-t} = \\dfrac{\\sin t}{t} \\ .\r\n\\]\r\n\u306a\u306e\u3067, [1] \u3092\u7528\u3044\u308c\u3070\r\n\\[\r\n\\displaystyle\\lim _ {x \\rightarrow -0} \\dfrac{\\sin x}{x} = \\displaystyle\\lim _ {t \\rightarrow +0} \\dfrac{\\sin t}{t} = 1 \\quad ... [2] \\ .\r\n\\]\r\n\u3088\u3063\u3066, [1] [2] \u3088\u308a\r\n\\[\r\n\\displaystyle\\lim _ {x \\rightarrow 0} \\dfrac{\\sin x}{x} = 1 \\ .\r\n\\]\r\n\u3053\u308c\u3092\u7528\u3044\u308c\u3070, \\(\\sin x\\) \u306e\u5c0e\u95a2\u6570 \\(( \\sin x )'\\) \u306f, \u5c0e\u95a2\u6570\u306e\u5b9a\u7fa9\u5f0f\u304b\u3089\r\n\\[\\begin{align}\r\n( \\sin x )' & = \\displaystyle\\lim _ {h \\rightarrow 0} \\dfrac{\\sin (x+h) -\\sin x}{h} \\\\\r\n& = \\displaystyle\\lim _ {h \\rightarrow 0} \\dfrac{2 \\cos \\left( x+\\frac{h}{2} \\right) \\sin \\frac{h}{2}}{h} \\\\\r\n& = \\displaystyle\\lim _ {h \\rightarrow 0} \\dfrac{\\sin \\frac{h}{2}}{\\frac{h}{2}} \\cdot \\cos \\left( x+\\dfrac{h}{2} \\right) \\\\\r\n& = 1 \\cdot \\cos x \\\\\r\n& = \\cos x \\ .\r\n\\end{align}\\]\r\n","protected":false},"excerpt":{"rendered":"\u4e09\u89d2\u95a2\u6570\u306e\u6975\u9650\u306b\u95a2\u3059\u308b\u516c\u5f0f \\[ \\displaystyle\\lim _ {x \\rightarrow 0} \\dfrac{\\sin x}{x} = 1 \\] \u3092\u793a\u3059\u3053\u3068\u306b\u3088\u308a, \\(\\sin x\\) \u306e\u5c0e\u95a2\u6570\u304c \\(\\ &hellip; <a href=\"https:\/\/www.roundown.net\/nyushi\/osr201301\/\">\u7d9a\u304d\u3092\u8aad\u3080 <span class=\"meta-nav\">&rarr;<\/span><\/a>","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"inline_featured_image":false,"footnotes":""},"categories":[90],"tags":[142,111],"class_list":["post-813","post","type-post","status-publish","format-standard","hentry","category-osaka_r_2013","tag-osaka_r","tag-111"],"_links":{"self":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/813","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/comments?post=813"}],"version-history":[{"count":0,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/813\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/media?parent=813"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/categories?post=813"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/tags?post=813"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}