当x→0时，lim（tanx－sinx）/x的极限怎么算 当x趋于0时，lim(tanx-sinx)/x³怎...
\u6c42X\u8d8b\u8fd1\u4e8e0\u65f6\uff0clim(tanx-sinx/x)\u3002\u5b8c\u5168\u53ef\u4ee5\u662f\u7528\u56db\u5219\u8fd0\u7b97\u505a\u7684 \u4f30\u8ba1LZ\u9898\u76ee\u6ca1\u770b\u6e05...
\u4f60\u4e24\u79cd\u65b9\u6cd5\u7b97\u7684\u4e0d\u662f\u4e00\u4e2a\u6781\u9650\u554a\uff01\uff01\u6f0f\u4e86\u4e00\u4e2a\u62ec\u53f7
\u7b2c\u4e00\u884c\u662f tanx-(sinx/x) \u5f53x\u63a5\u8fd10 tanx\u4e3a0 sinx/x\u4e3a1 \u7b54\u6848\u4e3a-1
\u7b2c\u4e8c\u884c\u662f (tanx-sinx)/x tanx/x\u4e3a1 sinx/x\u4e3a1 \u6240\u4ee5\u7b54\u6848\u4e3a0
\u7b2c\u4e00\u884c \u7528\u6d1b\u5fc5\u8fbe\u6cd5\u5219\u8ba1\u7b97\u7684\u8bdd =limx\u21920(sec^2x*x+tanx-cosx)=0+0-1=-1
\u7b2c\u4e8c\u884c\u7528\u6d1b\u5fc5\u8fbe\u6cd5\u5219\u8ba1\u7b97\u7684\u8bdd =limx\u21920(sec^2x-cosx)=1-1=0
原式=lim（sinx/cosx-sinx)/sin3x 约分 =lim(1/cosx-1)/sin2x =lim(1-cosx)/(sin2xcosx) 1-cosx~x2/2 sinx~x 所以原式=(x2/2)/(x2cosx) =1/(2cos0) =1/2
绛旓細lim(tanx-x)/x^3 =lim(secxsecx-1)/3x^2 =lim(2secxsecxtanx)/6x =1/3limsecxsecx =1/3 鍗lim锛坱anx-x锛=锛1/3锛墄^3銆傚緱璇併傛鎺ㄧ敤娉板嫆鍏紡锛歠(x)=tanx锛宖'(x)=(secx)^2锛宖''(x)=2(secx)^2tanx锛宖(3)(x)=4(secx)^2(tanx)^2+2(secx)^4 閭ｄ箞f(0)=0锛f'(0...
绛旓細绠鍗曞垎鏋愪竴涓嬶紝绛旀濡傚浘鎵绀
绛旓細lim(tanx-sinx)/sin³x =lim(sinx/cosx -sinx)/sin³x =lim(1/cosx -1)/sin²x =lim(1-cosx)/[cosx路(1-cos²x)]=lim(1-cosx)/[cosx路(1+cosx)(1-cosx)]=lim1/[cosx(1+cosx)]=1/[1脳(1+1)]=1/2 鏈闈炲父绠鍗曪紝杩炵瓑浠锋棤绌峰皬閮芥病鏈夌敤鍒帮紝閫氳繃涓...
绛旓細tanx-sinx=2tan(x/2)/(1+tan^2(x/2))-2tan(x/2)/(1+tan^2(x/2)),閫氬垎鍚庯紝寰楀嚭tanx-sinx=4tan^3(x/2)/(1-tan^4(x/2))褰搙瓒嬭繎浜0鏃讹紝tanx瓒嬭繎浜0銆傛晠褰搙瓒嬭繎浜0鏃讹紝lim(tanx-sinx)=0
绛旓細鍗lim锛坱anx-x锛= 锛1/3锛墄^3 寰楄瘉 姝ｆ帹鐢ㄦ嘲鍕掑叕寮忥細f(x)=tanx,f'(x)=(secx)^2,f''(x)=2(secx)^2tanx,f(3)(x)=4(secx)^2(tanx)^2+2(secx)^4 閭ｄ箞f(0)=0,f'(0)=1,f''(0)=0,f(3)(0)=2 tanx=0+x+0+(2/3!)x^3+o(x^3)=x+(1/3)x^3+o(x^3) ...
绛旓細lim(tanx-sinx)/(x^2*sinx)=limtanx(1-cosx)/(x^2*sinx)(绛変环鏃犵┓灏忎唬鎹)=limx(x^2/2)/(x^2*x)=1/2
绛旓細tanx-(sinx/x) 褰搙鎺ヨ繎0 tanx涓0 sinx/x涓1 绛旀涓-1 绗簩琛屾槸 (tanx-sinx)/x tanx/x涓1 sinx/x涓1 鎵浠ョ瓟妗堜负0 绗竴琛 鐢ㄦ礇蹇呰揪娉曞垯璁＄畻鐨勮瘽 =limx鈫0(sec^2x*x+tanx-cosx)=0+0-1=-1 绗簩琛岀敤娲涘繀杈炬硶鍒欒绠楃殑璇 =limx鈫0(sec^2x-cosx)=1-1=0 ...
绛旓細limx鈫0 (tanx-x)/x绔嬫柟 =limx鈫0 (sec骞虫柟x-1)/3x骞虫柟 =limx鈫0 (tan骞虫柟x)/3x骞虫柟 =limx鈫0 (x骞虫柟)/3x骞虫柟 =1/3 鎵浠 x鈫0鏃讹紝 tanx-x鏄痻鐨3闃舵棤绌峰皬銆
绛旓細绠鍗曞垎鏋愪竴涓嬶紝绛旀濡傚浘鎵绀
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 褰撳垎姣嶇瓑浜庨浂鏃讹紝灏变笉鑳藉皢瓒嬪悜鍊肩洿鎺ヤ唬鍏ュ垎姣嶏紝鍙互...
