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...

\u6c42\u6781\u9650\uff0c\u5f53x\u8d8b\u5411\u4e8e0\uff0clim(x-tanx)/(x^2*sinx),\u8981\u8fc7\u7a0b

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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

鏋侀檺鐨勮绠楅lim(x瓒嬩簬0)(tanx-x)/(x^3)绛旀绛変簬3,鍏蜂綋鎬庝箞绠..._鐧...
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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 ...

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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

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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

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