求X趋向于0时，lim(tanX-sinX)/(sin2X)^3 求极限(x趋向于0时）lim[sinx-sin(sinx)]...
\u5f53x\u8d8b\u4e8e0\u65f6\uff0clim\uff08tanx\uff0dsinx\uff09\u2571sin³x\u600e\u4e48\u6c42\uff1flim(tanx-sinx)/sin³x
=lim(sinx/cosx -sinx)/sin³x
=lim(1/cosx -1)/sin²x
=lim(1-cosx)/[cosx\u00b7(1-cos²x)]
=lim(1-cosx)/[cosx\u00b7(1+cosx)(1-cosx)]
=lim1/[cosx(1+cosx)]
=1/[1\u00d7(1+1)]
=1/2
\u672c\u9898\u975e\u5e38\u7b80\u5355\uff0c\u8fde\u7b49\u4ef7\u65e0\u7a77\u5c0f\u90fd\u6ca1\u6709\u7528\u5230\uff0c\u901a\u8fc7\u4e09\u89d2\u6052\u7b49\u53d8\u5f62\uff0c\u5c31\u53ef\u4ee5\u6c42\u51fa\u6781\u9650\u3002
0\u6bd40\u578b\u6781\u9650\uff0c\u8bf7\u7528\u6d1b\u5fc5\u8fbe\u6cd5\u5219\u3002\u5373\uff0c\u5206\u5f0f\u4e0a\u4e0b\u5206\u522b\u6c42\u5bfc\u3002
[sinx-sin(sinx)]\u2018=cosx-cosxcos(sinx),x\u21920\uff0c\u21921-1*1=0
(sinx)^3=3cosxsinx^2=0
\u7ee7\u7eed\u4f7f\u7528\u6d1b\u5fc5\u8fbe\u6cd5\u5219
\u3010cosx-cosxcos(sinx)\u3011'=sinx+sinxcos(sinx)+cosxcosxsin(sinx)=0
[3cosxsinx^2]'=-3sinx^3+6cosx^2*sinx=0
\u3010-sinx+sinxcos(sinx)+cosxcosxsin(sinx)\u3011=-cosx+cosxcos(sinx)-sinxcosxsin(sinx\uff09-2cosxsinxsin\uff08sinx)+2cosx^2*cosxcos(sinx)
N\u7684\u76f8\u5e94\u6027\u3000
\u4e00\u822c\u6765\u8bf4\uff0cN\u968f\u03b5\u7684\u53d8\u5c0f\u800c\u53d8\u5927\uff0c\u56e0\u6b64\u5e38\u628aN\u5199\u4f5cN(\u03b5)\uff0c\u4ee5\u5f3a\u8c03N\u5bf9\u03b5\u7684\u53d8\u5316\u800c\u53d8\u5316\u7684\u4f9d\u8d56\u6027\u3002\u4f46\u8fd9\u5e76\u4e0d\u610f\u5473\u7740N\u662f\u7531\u03b5\u552f\u4e00\u786e\u5b9a\u7684\uff1a\uff08\u6bd4\u5982\u82e5n>N\u4f7f|xn-a|N+1\u3001n>2N\u7b49\u4e5f\u4f7f|xn-a|<\u03b5\u6210\u7acb\uff09\u3002\u91cd\u8981\u7684\u662fN\u7684\u5b58\u5728\u6027\uff0c\u800c\u4e0d\u5728\u4e8e\u5176\u503c\u7684\u5927\u5c0f\u3002
(tanx-sinx)/(sin2x)^3
=(sinx/cosx-sinx)/(2sinxcosx)^3
=sinx(1-cosx)/[8(sinx)^3*(cosx)^4]
=(1-cosx)/[8(sinx)^2*(cosx)^4]
=2[sin(x/2)]^2/{8[2sin(x/2)cos(x/2)]^2*(cosx)^4}
=1/{16(cos(x/2)]^2*cosx)^4}
∴lim(x->0)(tanx-cosx)/(sin2x)^3
=1/lim(x->0){16[cos(x/2)]^2*(cosx)^4}
=1/(16*1^2*1^4)
=1/16.
绛旓細鎵浠^tan-e^x绛変环浜巘anx-x x鈫0鏃讹紝tanx-x绛変环浜巟^n锛=lim(x鈫0) (tanx-x)/x^n =lim(x鈫0) ((secx)^2-1)/nx^(n-1)=lim(x鈫0) (tanx)^2/nx^(n-1)=lim(x鈫0) x^2/nx^(n-1)=lim(x鈫0) x^(3-n)/n n=3 褰撳垎姣嶇瓑浜闆舵椂锛灏变笉鑳藉皢瓒嬪悜鍊肩洿鎺ヤ唬鍏ュ垎姣嶏紝鍙互...
绛旓細鍒掑浘鍍忥紝鏁板舰缁撳悎锛屾渶绠鍗曘傚洜涓篺(x)=tanx鍦▁=0鏃鏄繛缁嚱鏁,x瓒嬩簬0鍗充负tan(0)=0,鏋侀檺绗﹀彿鐪佺暐銆
绛旓細鏋侀檺=3*lim 銆愶紙sin3x/3x锛*锛1/cos3x锛夈=3
绛旓細lim(x鈫0锛塴n(tan7x)/ln(tan2x)锛漧im(x鈫0锛7sec^2(7x)/(tan7x)*(tan2x) /[2sec^2(2x)]鐢变簬sec2x锛漜os^2x鈫1锛屽綋x鈫0鏃 鎵浠 =1
绛旓細搴旂敤娲涘繀杈炬硶鍒欍傚綋x瓒嬭繎浜0鏃讹紝lim(arctan x)/x=lima(arctan x)'/x'=lim1/(x^2+1)=1銆備护arctanx=t lim(arctanx/x)=lim(t/tant)=lim锛坱/sint锛*lim cost=1 鎵浠rctanx~x 搴旂敤鏉′欢 鍦ㄨ繍鐢ㄦ礇蹇呰揪娉曞垯涔嬪墠锛岄鍏堣瀹屾垚涓ら」浠诲姟锛氫竴鏄垎瀛愬垎姣嶇殑鏋侀檺鏄惁閮界瓑浜庨浂(鎴栬呮棤绌峰ぇ)锛...
绛旓細x瓒嬭繎浜0鏃,璁＄畻limsinx锛1+x)/tan2x,瑕佽绠楄繖涓瓑寮,鎴戜滑瑕佺煡閬2涓瓑浠锋棤绌峰皬,x瓒嬭繎浜0鏃,sinx~x,tanx~x,鎵浠,x瓒嬭繎浜0鏃,limsinx锛1+x)/tan2x=Lim x(1+x)/2x =1/2
绛旓細鍏蜂綋鍥炵瓟濡備笅锛lim(x瓒嬪悜浜0+锛墄^tanx =e^lim(x瓒嬪悜浜0+锛塴nx^tanx =e^lim(x瓒嬪悜浜0+锛塴nx*tanx =e^lim(x瓒嬪悜浜0+锛塴nx/cotx (鈭/鈭)=e^lim(x瓒嬪悜浜0+锛(1/x)/(-csc^2x)=e^lim(x瓒嬪悜浜0+锛-sinx =e^0 =1 鏋侀檺鍑芥暟鐨勬剰涔夛細鍦ㄥ尯闂(a-蔚锛宎+蔚)涔嬪鑷冲鍙湁N涓紙鏈夐檺涓...
绛旓細x瓒嬩簬0锛閭ｄ箞鍒嗗瓙鍒嗘瘝閮芥槸瓒嬩簬0鐨 鍒嗘瘝sinbx绛変环浜巄x 浣跨敤娲涘繀杈炬硶鍒 鍒嗗瓙鍒嗘瘝鍚屾椂姹傚 鍘熸瀬闄=limx鈫0 lntan(蟺/4+ax)/bx =limx鈫0 tan(蟺/4+ax) *1/cos²(蟺/4+ax) *a/b =1/(1/2) *a/b=2a/b
绛旓細杩欎釜寰堟槑鏄惧憖锛寈瓒嬪悜浜0+锛宼anx瓒嬪悜浜0+锛宭n锛坱anx锛夎秼杩戜簬璐熸棤绌凤紝鐢讳釜鍥惧氨鐪嬪埌浜...
绛旓細缁撴灉灏辨槸1锛岃繃绋嬪涓嬶細
