当x趋向于0时，xsin1/x的极限存在吗？为啥有人算的不存在 还有根据无穷 微积分高手请进，为什么当x趋近于0时，sin(xsin(1/...
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绛旓細褰搙瓒嬭繎浜0鏃讹紝鎺㈣x涔樹互sin(1/x)鐨勬瀬闄愰棶棰橈紝鎴戜滑闇瑕佹敞鎰忕殑鏄紝鐢变簬1/x瓒嬪悜浜庢棤绌峰ぇ锛宻in(1/x)骞朵笉鑳界瓑鍚屼簬1/x鐨勬瀬闄愯涓恒傜浉鍙嶏紝鎴戜滑鍙互杩愮敤绛変环鏃犵┓灏忕殑鎬濇兂锛屽嵆褰搙鎺ヨ繎0鏃讹紝sinx涓巟杩戜技鐩哥瓑銆傚悓鏃讹紝褰搙瓒嬩簬鏃犵┓澶ф椂锛1/x浼氳秼杩戜簬0锛屾墍浠in(1/x)鐨勬瀬闄愬彲浠ユ浛鎹负1/x銆傛瀬闄愮殑姹傝В...
绛旓細x鈫0鏃,limx鏄棤绌峰皬,sin1/x涓烘湁鐣岄噺.鍥犳涓よ呬箣绉槸鏃犵┓灏忛噺=0.鏈夌晫閲忎箻浠ユ棤绌峰皬閲忎粛鏄棤绌峰皬.鏃犵┓灏忛噺鏄暟瀛﹀垎鏋愪腑鐨勪竴涓蹇碉紝鐢ㄤ互涓ユ牸鍦板畾涔夎濡傗滄渶缁堜細娑堝け鐨勯噺鈥濄佲滅粷瀵瑰兼瘮浠讳綍姝ｆ暟閮借灏忕殑閲忊濈瓑闈炴寮忔弿杩般
绛旓細鏃犵┓灏忎笌鏈夌晫鍑芥暟鐨勪箻绉槸鏃犵┓灏.涔熷彲浠ヨ繖鏍风悊瑙ｏ細|sin(1/x)|<=1锛鍥犳 |x*sin(1/x)|<=|x|,鍥犳lim |x*sin(1/x)| <= lim |x|=0;鍥犳lim x*sin(1/x)=0
绛旓細浣犲ソ銆傝繖涓槸鏃犵┓灏忎箻浠ユ湁鐣屽嚱鏁般傚洜涓sin1/x鏄釜鏈夌晫鍑芥暟锛屽湪[-1,1]涔嬮棿锛屾墍浠ヤ箻浠ヤ竴涓瓒嬭繎0鐨勬暟锛岃繕鏄=0 鏋侀檺=0
绛旓細鎭掓湁|y锛0|锛潀xsin(1/x)锛0|锛溛 鎵浠ワ紝y锛漻sin(1/x) 褰搙鈫0鏃涓烘棤绌峰皬 鍊嶈鍗婅鍏紡锛歴in ( 2伪 ) = 2sin伪 路 cos伪 sin ( 3伪 ) = 3sin伪 - 4sin & sup3 ; ( 伪 ) = 4sin伪 路 sin ( 60 + 伪 ) sin ( 60 - 伪 )sin ( 伪 / 2 ) = 卤 鈭( ( 1 -...
绛旓細y=xsin(1/x)涓簒鈫0鏃鐨勬棤绌峰皬锛堟敞鎰忥紝涓嶆槸鏃犵┓澶э紒锛侊級lim(x鈫0)y =lim x*sin(1/x)鍥犱负锛宻in(1/x)涓烘湁鐣岄噺锛x瓒嬩簬0锛屼负 鏃犵┓灏忛噺 鏈夌晫閲忎笌鏃犵┓灏忛噺鐨勭Н鏄棤绌峰皬閲 鍥犳锛=lim x*sin(1/x)=0 鏈変笉鎳傛杩庤拷闂
绛旓細鏄釜鏃犵┓灏忋傝宻in锛1/x锛夋槸鏈夌晫鍑芥暟銆傛牴鎹湁鐣屽嚱鏁板拰鏃犵┓灏忕浉涔橈紝缁撴灉杩樻槸鏃犵┓灏忕殑瀹氱悊銆傛墍浠褰搙鈫0+鐨勬椂鍊欙紝xsin锛1/x锛夎繕鏄棤绌峰皬锛屾瀬闄愭槸0鑰屼笉鏄1銆傝嫢鏁板垪鐨勬瀬闄愬瓨鍦紝鍒欐瀬闄愬兼槸鍞竴鐨勶紝涓斿畠鐨勪换浣曞瓙鍒楃殑鏋侀檺涓庡師鏁板垪鐨勭浉绛夈傚鏋滀竴涓暟鍒楁敹鏁涳紙鏈夋瀬闄愶級锛岄偅涔堣繖涓暟鍒椾竴瀹氭湁鐣屻
绛旓細x瓒嬩簬0 -lim|x|<=limxsin1/x<=lim|x| 0<=|limxsin1/x|<=0 鈭 limxsin1/x=0
绛旓細sin(xsin1/x锛夋瘮涓妜sin1/x 鍙互鐢ㄥ叕寮忥紙褰揳瓒嬩簬鏃犵┓灏鐨勬椂鍊欙紝鏈変竴涓叕寮(sin a)/a=1锛夈傜瓟妗堢瓑浜1銆傛瀬闄愮殑姹傛硶鏈夊緢澶氱锛1銆佽繛缁垵绛夊嚱鏁帮紝鍦ㄥ畾涔夊煙鑼冨洿鍐呮眰鏋侀檺锛屽彲浠ュ皢璇ョ偣鐩存帴浠ｅ叆寰楁瀬闄愬硷紝鍥犱负杩炵画鍑芥暟鐨勬瀬闄愬煎氨绛変簬鍦ㄨ鐐圭殑鍑芥暟鍊笺2銆佸埄鐢ㄦ亽绛夊彉褰㈡秷鍘婚浂鍥犲瓙锛堥拡瀵浜0/0鍨嬶級銆3銆...
绛旓細涓嶆槸鐨勶紝鏋侀檺涓嶅瓨鍦ㄣ傚湪x-->0杩囩▼涓锛寈sin1/x 鍙彇鍒板緱0鐐癸紝涔熷嵆鎵句笉鍒颁换浣曚竴涓幓蹇冮偦鍩烾(x,未)浣垮緱鍒嗘瘝鏈夋剰涔夛紝鏁呮瀬闄愪笉瀛樺湪銆傛暟瀛︽槸浜虹被瀵逛簨鐗╃殑鎶借薄缁撴瀯涓庢ā寮忚繘琛屼弗鏍兼弿杩扮殑涓绉嶉氱敤鎵嬫锛屽彲浠ュ簲鐢ㄤ簬鐜板疄涓栫晫鐨勪换浣曢棶棰橈紝鎵鏈夌殑鏁板瀵硅薄鏈川涓婇兘鏄汉涓哄畾涔夌殑銆備粠杩欎釜鎰忎箟涓婏紝鏁板灞炰簬褰㈠紡绉戝...
