当x趋近0,sinx除以x的极限为什么等于1?不理解 为什么sinx/x的极限等于1(当x趋向于0)?
\u5f53x\u8d8b\u8fd10\uff0csinx\u9664\u4ee5x\u4e3a\u4ec0\u4e48\u7b49\u4e8e1\uff1f\u4e0d\u7406\u89e3\u5c06sinx\u8fdb\u884c\u6cf0\u52d2\u5c55\u5f00\u5f97\u5230\uff0csinx=x-x^3/3!+x^5/5!-x^7/7!+\u2026\u2026
\u5f53x\u8d8b\u8fd1\u4e8e0\u7684\u65f6\u5019\uff0c\u4ece\u5c55\u5f00\u5f0f\u7684\u7b2c\u4e8c\u9879\u5f00\u59cb\u5747\u4e3ax\u7684\u9ad8\u9636\u65e0\u7a77\u5c0f\u91cf\uff0c\u53ef\u4ee5\u5ffd\u7565\uff0c\u6240\u4ee5
sinx\u2248x,\u6240\u4ee5\u6781\u9650lim
x/sinx=1
\u56e0\u4e3asinx<x<tanx,
\u6240\u4ee5
1/sinx>1/x>cosx/sinx
绛旓細sinx/x锛褰搙鈫0鐨勬椂鍊欐瀬闄愭槸1,sinx/x锛屽綋x鈫掆垶鐨勬椂鍊欐瀬闄愭槸0锛鍥犱负褰搙瓒嬭繎浜庢棤绌峰ぇ鐨勬椂鍊檚inx鐨勫彇鍊艰寖鍥存槸[-1,1]銆傝寈涓哄垎姣嶏紝褰撹秼杩浜庢棤绌峰ぇ鐨勬椂鍊檚inx/x鐨鏋侀檺鏄0銆傛棤闄愰潬杩戣屾案杩滀笉鑳藉埌杈锯濈殑鎰忔濄傛暟瀛︿腑鐨勨滄瀬闄愨濇寚锛氭煇涓涓嚱鏁颁腑鐨勬煇涓涓彉閲忥紝姝ゅ彉閲忓湪鍙樺ぇ鐨勬案杩滃彉鍖栫殑杩囩▼涓紝閫愭笎...
绛旓細0 鈮 lim(胃鈫0) (胃/sin胃) 鈮 1 鐢变簬胃/sin胃鍦ㄎ歌秼杩戜簬0鏃舵槸杩炵画鐨勶紝鏍规嵁澶归煎畾鐞嗭紝鏋侀檺鍊煎繀椤昏惤鍦ㄨ繖涓や釜杈圭晫鐨勫叕鍏遍儴鍒嗭紝鍗1銆傚洜姝わ紝鎴戜滑寰楀嚭缁撹锛歭im(胃鈫0) (胃/sin胃) = 1 杩欏氨鏄负浠涔褰揦瓒嬪悜浜0鏃锛宻inX / X鐨鏋侀檺绛変簬1鐨勬暟瀛﹁В閲娿傝繖涓湅浼肩畝鍗曠殑鏋侀檺闂锛屽叾瀹炶暣鍚潃娣卞埢...
绛旓細瑙i杩囩▼濡備笅锛歭imsinx锛坸->0锛=0 limx锛坸->0锛=0 锛坰inx锛'=cosx锛(x)'=1 =lim(sinx/x)=lim(cosx/1)=cos0 =1
绛旓細x瓒嬭繎浜0鏃,sinx鍒嗕箣涓鐨勬瀬闄愬涓 锛1銆褰 x鈫0鏃讹紝sin(1/x) 鐨勫煎湪[-1,1]鍐呮尝鍔紝鏋侀檺褰撶劧涓嶅瓨鍦 2銆佽 x*sin(1/x) 鏄剧劧鏄瓒嬩簬0鐨 鈥滄瀬闄愨濇槸鏁板涓殑鍒嗘敮鈥斺斿井绉垎鐨勫熀纭姒傚康銆傚箍涔夌殑鈥滄瀬闄愨濇槸鎸団滄棤闄愰潬杩戣屾案杩滀笉鑳藉埌杈锯濈殑鎰忔濄傛暟瀛︿腑鐨勨滄瀬闄愨濇寚锛氭煇涓涓嚱鏁颁腑鐨勬煇涓涓...
绛旓細褰搙瓒嬭繎浜0鏃,鐢辨棤绌峰皬鍙煡锛宻inx=x锛屾墍浠ュ師寮忔瀬闄愪负1 婊℃剰璇烽噰绾
绛旓細1
绛旓細杩欐槸瑕佺湅x鐨鍙栧艰寖鍥寸殑銆傚垎涓轰互涓嬩袱绉嶆儏鍐点1锛褰搙鏃犻檺瓒嬭繎浜0鏄锛宻inx/x=1 杩欐槸楂樼瓑鏁板涔︿笂鐨勫畾鐞嗐2锛氳屽綋x鏃犻檺瓒嬭繎浜庢棤绌风殑鏃跺欙紝sinx/x=0.杩欎釜鏃跺欏彲浠ユ妸x褰撳仛鏃犵┓灏忕殑涓涓暟锛岃宻inx鏄湁鐣屽嚱鏁帮紝鍏惰寖鍥翠负銆-1--1銆戙傚浘涓涓烘寮﹀嚱鏁帮紝鍥句簩涓轰綑寮﹀嚱鏁般傛棤绌峰皬鐨勫嚱鏁*鏈夌晫鍑芥暟锛岀粨鏋滆嚜鐒...
绛旓細鐢眑im锛坸鈫0锛塩osx=1鍙婂嚱鏁版瀬闄愮殑杩暃鎬э紝鍗冲緱lim锛坸鈫0锛sinx/x=1銆俵im锛坸鈫0锛夋槸鎸x瓒嬭繎浜0鐨鏋侀檺銆備互 鐨勬瀬闄愪负渚嬶紝f(x) 鍦ㄧ偣 浠涓烘瀬闄愮殑瀹氫箟鏄細 瀵逛簬浠绘剰缁欏畾鐨勬鏁拔碉紙鏃犺瀹冨涔堝皬锛夛紝鎬诲瓨鍦ㄦ鏁 锛屼娇寰褰搙婊¤冻涓嶇瓑寮 鏃讹紝瀵瑰簲鐨勫嚱鏁板糵(x)閮芥弧瓒充笉绛夊紡锛 锛岄偅涔堝父鏁癆...
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绛旓細鏄殑,褰搙瓒嬪悜0鏃,x绛変环浜sinx