X趋向于0时,tanX-sinX除于In(1+x^3 的极限
tanx-sinx=tanx(1-cosx)=tanx•2(sinx/2)^2,利用等价无穷小：tanx～x,sinx～x,ln(1+x)～x化简原式=2(x•x^2/4)/x^3=1/2
绛旓細sinx=2sin(x/2)cos(x/2)/(sin^2(x/2)+cos^2(x/2)),鍒嗗瓙鍒嗘瘝鍚岄櫎浠os^2(x/2),寰楀埌sinx=2tan(x/2)/(1+tan^2(x/2))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...
绛旓細鍦x瓒嬩簬0鐨勬椂鍊 tanx~sinx~e^x-1~ln(1+x)锛岄兘鏄瓑浠蜂簬x鐨 鑰1-cosx~1/2x²锛岃繖閲岀殑1銆乴im x鈫0锛(tanx-sinx)/(sin2x)^3 =lim x鈫0锛宼anx *(1-cosx)/(sin2x)^3 浠ｅ叆寰楀埌 绛変环浜巐im x鈫0 (x *1/2x²) /(2x)³ =1/16 lim x鈫0锛宻in(x-1)tan(x-1...
绛旓細tanx sin3x 鍦0澶 鍒嗗埆娉板嫆灞曞紑 绗竴椤瑰垎鍒槸 x 鍜 3x 鍥犱负瀹冧滑姝ゆ椂鍒嗗埆鏄痻鐨勫悓闃舵棤绌峰皬 鑰屼笖 鐩稿噺鍚 涓嶄細鎶垫秷 寰楀埌-2x 浠荤劧鏄拰 x 鍚岄樁鏃犵┓灏 鍥犳 鍙互鍜 x鍚岄樁 鍜-2x 绛変环鏃犵┓灏
绛旓細tan(tanx) -sin(sinx)=[x+(2/3)x^3 +o(x^3) ] -[x-(1/3)x^3+o(x^3)]=x^3 +o(x^3)tan(tanx) -sin(sinx) 绛変环浜 x^3 lim(x->0) [tan(tanx) -sin(sinx)]/(tanx-sinx)甯﹀叆浠ヤ笂绛変环 =lim(x->0) x^3/[ (1/2)x^3]=2 ...
绛旓細鍏蜂綋鍥炵瓟濡備笅锛tanx -sinx =tanx-tanx路cosx =tanx(1-cosx)~x路(x² /2)=x³/2 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...
绛旓細杩欎釜鏄珮绛夋暟瀛﹂噷闈㈢殑姹傛瀬闄愰棶棰橈紝绠楁槸鍩烘湰棰樼洰锛岀粰浣犱竴涓В棰樻濊矾銆傛妸limx->0鏃讹紝tanx=x,sinx=x,杩欐牱涓婇潰鐨勫紡瀛愬氨鏄紝(tanx-sinx)/x^3锛岀劧鍚庢妸tanx鍒嗚В鎴恈osx鍜宻inx锛屽啀娆″寲绠锛屾渶鍚庢湁涓(1-cosx)杩欎釜寮忓瓙鏄-1/2*x^2杩樻槸1/2*x^2锛岃涓嶆竻浜嗭紝鑷繁鍋氫竴鍋氬氨鐭ラ亾浜嗭紝杩欑楂樼瓑鏁板鐨勫熀鏈鐩...
绛旓細鍘熷紡=lim锛坰inx/cosx-sinx)/sin3x 绾﹀垎 =lim(1/cosx-1)/sin2x =lim(1-cosx)/(sin2xcosx) 1-cosx~x2/2 sinx~x 鎵浠ュ師寮=(x2/2)/(x2cosx) =1/(2cos0) =1/2
绛旓細=lim[2sec(tanx) * sec^2 (tan(tanx)) +sin(sinx) * cosx] / 6x (褰搙鈫0鏃)鍚屼笂锛屽綋x鈫0鏃讹紝sec^2 (x)鈫1锛宑osx鈫1锛宑os^2 (x)鈫1 浠ｅ叆骞跺寲绠鏁寸悊寰楀埌锛=lim[2sin(tanx) + sin(sinx)]/6x (褰搙鈫0鏃) 鍐嶅鏈畾寮忚繍鐢ㄦ礇蹇呰揪娉曞垯锛=lim[2cos(tanx) * sec^2 ...
绛旓細=(1/cosx-1)/x^2 =(1-cosx)/x^2 =2*sin^2(x/2)/x^2 =1/2 lim鐪佺暐浜 浣犵殑閭ｄ釜鎵璋撶瓟妗堣偗瀹氶敊浜嗭紝浣犳兂锛屽鏋渪-->0+,閭ｄ箞tanx鏄洿瑙掕竟鐨勬瘮锛岃宻in鏄洿瑙掕竟姣旀枩杈癸紝鍒嗗瓙褰撶劧>0,鍥犳缁撴灉鑲畾>=0鐨
绛旓細lim (tanx-sinx)/sin³x x鈫0 =lim (sinx/cosx -sinx)/sin³x x鈫0 =lim (1-cosx)/(sin²xcosx)x鈫0 =lim ½x²/(x²路cos0)x鈫0 =½
