当x→0时，求tanx-sinx关于x的阶数，如何知道它的阶数？ 当x趋于0时，如何求无穷小量tanx-sinx关于x的阶
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lim(tanx-sinx)/(x^2*sinx)=limtanx(1-cosx)/(x^2*sinx)(等价无穷小代换)=limx(x^2/2)/(x^2*x)=1/2绛旓細=limx鈫0+sin(tanx)tan(sinx) 锛堢畻鍑洪潪闆舵瀬闄愶級=limx鈫0+cos(sinx)sec2x2sin(tanx)sec2(sinx)cosx2tan(sinx) 锛堢敤缃楁瘮濉旀硶鍒欙級=limx鈫0+cos(sinx)sec2xsec2(sinx)cosx路limx鈫0+tan(sinx)sin(tanx)=limx鈫0+tan(sinx)sin(tanx)鍑虹幇寰幆锛屾鏃剁敤缃楁瘮濉旀硶鍒欐眰涓嶅嚭缁撴灉銆傛庝箞鍔烇紵
绛旓細lim x鈫0锛(tanx-sinx)/(sin2x)^3 搴旂敤缃楁瘮杈炬硶鍒欙紝鍒嗗瓙鍒嗘瘝鍚屾椂姹傚 涓婂紡=lim x鈫0锛(1/(cosx)^2-cosx)/[3*锛坰in2x)^2*cos2x*2]=1/6lim x鈫0(1-(cosx)^3)/(sin2x)^2 鍐嶆搴旂敤缃楁瘮杈炬硶鍒欙紝鍒嗗瓙鍒嗘瘝鍚屾椂姹傚 涓婂紡=1/6lim x鈫0锛3(cosx)^2*sinx)/2sin2x*cos2x*2)=1/8...
绛旓細1-cos2x=2sin^2 x 鍒欏師寮忓寲绠涓:[(1/cosx)-1]/2sinx =(1-cosx)/(2sinxcosx)=(1-cosx)/(sin2x)娲涘繀杈炬硶鍒欎笂涓嬫眰瀵硷細sinx/2cos2x 鍒欐瀬闄愪负0 楠岀畻浜嗗緢澶氭浜...搴旇鏄瓟妗堥敊浜嗗惂锛
绛旓細x-->0 lim ( tanx-sinx)/x =lim ( sinx-sinxcosx)/(xcosx)=lim (1-cosx)/cosx =0
绛旓細(tanx-sinx)/sin3x =(sinx/cosx-sinx)/sin3x =(1/cosx-1)/sin2x =[(1-cosx)/cosx]/(1-cos2x)=1/[cosx(1+cosx)]鎵浠ユ瀬闄=1/[1*(1+1)]=1/2
绛旓細=(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} 鈭磍im(x->0)(tanx-cosx)/(sin2x)^3 =1/lim(x->0){16[...
绛旓細瑙ｇ瓟杩囩▼涓猴細鐢辨嘲鍕掑叕寮忓彲寰楋細tanx=x+x^3/3+o(x^3)sinx=x-x^3/6+o(x^3)鍒tanx-sinx=x+x^3/3+o(x^3) -锛坸-x^3/6+o(x^3)锛=x^3/2銆傛墍浠inx-tanx鐨勭瓑浠锋棤绌峰皬涓簒^3/2銆傜瓑浠锋棤绌峰皬鏇挎崲鏄绠楁湭瀹氬瀷鏋侀檺鐨勫父鐢ㄦ柟娉曪紝瀹冨彲浠ヤ娇姹傛瀬闄愰棶棰樺寲绻佷负绠锛屽寲闅句负鏄撱
绛旓細tanx-sinx=tanx*(1-cosx)閭ｄ箞x瓒嬩簬0鐨勬椂鍊 tanx绛変环浜x锛1-cosx绛変环浜0.5x²鑰屽垎姣(sin2x)³绛変环浜(2x)³鎵浠ュ師鏋侀檺=lim(x瓒嬩簬0) 0.5x³/8x³=1/16
绛旓細0/0鍨嬬殑鏋侀檺涓嶈兘闅忎究鎷嗛」锛屽洜涓鸿繖鏍峰彲鑳介犳垚涓婁笅鏃犵┓灏忕殑闃跺彂鐢熷彉鍖栥俵im銆x鈫0銆曪紙tanx-sinx锛/x²锛漧im銆攛鈫0銆曪紙1-cosx锛塻inx/x²cosx 锛漧im銆攛鈫0銆曪紙sin²x锛塻inx/x²cosx锛1+cosx锛夛紳0/2 锛0
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...
