lim(x趋向于0)((tanx-sinx)/(x*(sinx)^2)) 求极限,要过程 高数题 lim(x趋于0)(tanx-sinx)/x^3的极...
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\u539f\u5f0f =lim(1-secx)/(2xsinx+x^2*cosx)
=lim(-2sinx/cos^3x)/(2sinx+2xcosx+2xcosx-x^2sinx)
=lim(-2-4sin^2x/cos^4x)/(2cosx+4cosx-4xsinx-2xsinx-x^2cosx)
=-1/3
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LIM(X\u8d8b\u54110\uff09tanx-sinx/x3\u6b21\u65b9=lim(x->0)tanx(1-cosx)/x³=lim(x->0)(x\u00b7x²/2)/x³=1/2
用泰勒公式和高阶无穷小:因为:tanx=x+x^3/3+o(x^3),sinx=x-x^3/3+0(x^3),tanx-sinx=x^3/3+x^3/3!+0(x^3)
(sinx)^2与x^2是等价无穷小
所以:lim(x趋向于0)((tanx-sinx)/(x*(sinx)^2))
=lim(x趋向于0)(x^3/3+x^3/3!+o(x^3))/x^3=1/2
lim(x趋向于0)((tanx-sinx)/(x*(sinx)^2)) =lim(x趋向于0)[(sinx/cosx-sinx)]/x(sinx)^2
=lim(x趋向于0)[1-cosx)/x(sinx)=lim(x趋向于0)2(sinx/2)^2/x2sinx/2cosx/2
=1/2lim(x趋向于0)[sinx/2/x/2)(1/cosx/2=lim(x趋向于0)1/cosx/2=1/2
两个重要极限 lim(x趋向于0)sinx/x=1
绛旓細绠鍗曞垎鏋愪竴涓嬶紝绛旀濡傚浘鎵绀
绛旓細lim(x瓒嬭繎浜0)(tan(tanx)-sin(sinx))/(x-sinx)=lim(x瓒嬭繎浜0)[(x+x^3/3+(x+x^3/3)^3/3)+o(x^3)]-[x-x^3/6-(x-x^3/6)^3+o(x^3)]/[x-(x-x^3/6+o(x^3)]=[(2/3)+(1/3)]/(1/6)=6 婊℃剰璇烽噰绾炽
绛旓細鎴戠殑绛旀 lim[x鈫0] (tanx-sinx)/x³=lim[x鈫0] (sinx/cosx-sinx)/x³=lim[x鈫0] (sinx-sinxcosx)/(x³cosx)=lim[x鈫0] sinx(1-cosx)/(x³cosx)=lim[x鈫0] sin³x(1-cosx)/(x³sin²xcosx)=lim[x鈫0] (sinx/x)³路(1-co...
绛旓細(tanx-x)'=sec²x-1 (x-sinx)'=1-cosx 褰搙鈫0鏃讹紝浠嶆槸0/0鍨嬶紝浣跨敤娲涘繀杈炬硶鍒 (sec²x-1)'=2secx*tanxsecx=2sinx/cos³x (1-cosx)'=sinx 鎵浠ュ師寮 =lim x鈫0[(2sinx/cos³x)/sinx]=lim x鈫0[(2/cos³x)]=2/1 =2 ...
绛旓細鍏蜂綋鍥炵瓟濡備笅锛tanx=sinx/cosx 褰搙鈫0 tanx =sinx =x 鍜岃鍏紡锛歴in ( 伪 卤 尾 ) = sin伪 路 cos尾 卤 cos伪 路 sin尾 sin ( 伪 + 尾 + 纬 ) = sin伪 路 cos尾 路 cos纬 + cos伪 路 sin尾 路 cos纬 + cos伪 路 cos尾 路 sin纬 - sin伪 路 sin尾 路 sin纬 cos (...
绛旓細鍦x瓒嬩簬0鐨勬椂鍊欙紝tanx鏄瓑浠蜂簬x鐨勩傛墍浠lim(x-0)(tanx-x)鐨勬瀬闄愭槸0銆
绛旓細鍒╃敤绛変环鏃犵┓灏忔潵璁$畻锛屽洜涓哄綋x瓒嬪悜0鏃讹紝tan(2x)~2x tan(4x)~4x 鎵浠ュ師寮=2x/4x=1/2
绛旓細lim (tan(tanx)-tanx)/(x³/2锛夊啀浠 f(x)=tanx 鍒檖1鐨勫垎瀛愭槸 f(tanx)-f(x)=f'(c)(tanx -x)(杩欓噷鐢ㄤ簡涓煎畾鐞嗭紝c鍦▁涓巘anx涔嬮棿)褰 x鈫0鏃讹紝c鈫0锛宖'(c)=sec²c鈫1 p1 = lim (tanx-x)/ (x³/2)=2/3 p2 = lim (tanx -sinx)/(tanx - sinx)...
绛旓細娉曚竴锛氭敼鍐欐垚姝d綑寮﹀嚱鏁板悗锛岀粨鍚堣繛缁ф眰鏋侀檺锛涙硶浜岋細鍊熷姪鍥惧儚锛岃瀵熷乏鍙虫瀬闄愬緱涔嬶紱娉曚笁锛氱敱浜庢鍒囧嚱鏁板湪鈥0鈥濈殑闄勮繎鏄繛缁殑锛岀敱杩炵画鎬х洿鎺ュ皢x=0浠e叆tanx寰楁瀬闄愬硷紱4.鍓嶄袱绉嶆柟娉曞鍥炬墍绀猴細
绛旓細缁撴灉涓猴細鏋侀檺鍊艰秼浜1/2 瑙i杩囩▼濡備笅锛歺瓒嬩簬0锛宑ot2x=1/tan2x绛変环浜1/2x 閭d箞姝ゆ瀬闄愬=lim(x瓒嬩簬0) x/tan2x =lim(x瓒嬩簬0) x/2x = 1/2 鈭存瀬闄愬艰秼浜1/2