limx趋近于01-cosx/x=多少 limx趋向于0 （1-cosx）/x²
limx\u8d8b\u8fd1\u4e8e01-cosx/x=\u591a\u5c11\u6781\u9650\u662f0\uff0c\u5982\u4e0b
limx\u8d8b\u5411\u4e8e0 \uff081-cosx\uff09/x²\u7684\u7ed3\u679c\u7b49\u4e8e1/2\u3002
\u89e3\uff1alim(x\u21920)(1-cosx)/(x^2) \uff08\u6d1b\u5fc5\u8fbe\u6cd5\u5219\uff0c\u5206\u5b50\u5206\u6bcd\u540c\u65f6\u6c42\u5bfc\uff09
=lim(x\u21920)sinx/(2x) \uff08\u6d1b\u5fc5\u8fbe\u6cd5\u5219\uff0c\u5206\u5b50\u5206\u6bcd\u540c\u65f6\u6c42\u5bfc\uff09
=lim(x\u21920)cosx/2
=1/2
\u5373lim(x\u21920)(1-cosx)/(x^2)\u7684\u6781\u9650\u503c\u7b49\u4e8e1/2\u3002
\u6269\u5c55\u8d44\u6599\uff1a
1\u3001\u6781\u9650\u8fd0\u7b97\u6cd5\u5219
\u4ee4limf(x)\uff0climg(x)\u5b58\u5728\uff0c\u4e14\u4ee4limf(x)=A\uff0climg(x)=B\uff0c\u90a3\u4e48
\uff081\uff09\u52a0\u51cf\u8fd0\u7b97\u6cd5\u5219
lim(f(x)\u00b1g(x))=A\u00b1B
\uff082\uff09\u4e58\u6570\u8fd0\u7b97\u6cd5\u5219
lim(a*f(x))=a*limf(x)\uff0c\u5176\u4e2da\u4e3a\u5df2\u77e5\u7684\u5e38\u6570\u3002
2\u3001\u6c42\u6781\u9650\u7684\u65b9\u6cd5
\uff081\uff09\u5206\u5b50\u5206\u6bcd\u6709\u7406\u5316
\uff082\uff09\u5939\u903c\u6cd5\u5219
3\u3001\u6781\u9650\u7684\u91cd\u8981\u516c\u5f0f
\uff081\uff09lim(x\u21920)sinx/x=1\uff0c\u56e0\u6b64\u5f53x\u8d8b\u4e8e0\u65f6\uff0csinx\u7b49\u4ef7\u4e8ex\u3002
\uff082\uff09lim(x\u21920)(1+x)^(1/x)=e\uff0c\u6216\u8005lim(x\u2192\u221e)(1+1/x)^x=e\u3002
\uff083\uff09lim(x\u21920)(e^x-1)/x=1\uff0c\u56e0\u6b64\u5f53x\u8d8b\u4e8e0\u65f6\uff0ce^x-1\u7b49\u4ef7\u4e8ex\u3002
\u53c2\u8003\u8d44\u6599\u6765\u6e90\uff1a\u767e\u5ea6\u767e\u79d1-\u6781\u9650
绛旓細lim(1-cosx)/x^2(x瓒嬩簬0锛=1/2銆傝В绛旇繃绋嬪涓嬶細鈥滄瀬闄愨濇槸鏁板涓殑鍒嗘敮鈥斺斿井绉垎鐨勫熀纭姒傚康锛屽箍涔夌殑鈥滄瀬闄愨濇槸鎸団滄棤闄愰潬杩戣屾案杩滀笉鑳藉埌杈锯濈殑鎰忔濄傛暟瀛︿腑鐨勨滄瀬闄愨濇寚锛氭煇涓涓嚱鏁颁腑鐨勬煇涓涓彉閲忥紝姝ゅ彉閲忓湪鍙樺ぇ锛堟垨鑰呭彉灏忥級鐨勬案杩滃彉鍖栫殑杩囩▼涓傞愭笎鍚戞煇涓涓‘瀹氱殑鏁板糀涓嶆柇鍦伴艰繎鑰屸滄案...
绛旓細绛旀鏄 0 褰x瓒嬪悜0鏃讹紝cosx瓒嬪悜1锛1-1=0
绛旓細limx瓒嬪悜浜0 锛1-cosx锛/x²鐨勭粨鏋滅瓑浜1/2銆傝В锛歭im(x鈫0)(1-cosx)/(x^2) 锛堟礇蹇呰揪娉曞垯锛屽垎瀛愬垎姣嶅悓鏃舵眰瀵硷級=lim(x鈫0)sinx/(2x) 锛堟礇蹇呰揪娉曞垯锛屽垎瀛愬垎姣嶅悓鏃舵眰瀵硷級=lim(x鈫0)cosx/2 =1/2 鍗砽im(x鈫0)(1-cosx)/(x^2)鐨勬瀬闄愬肩瓑浜1/2銆
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绛旓細0<=1-cosx=2sin²x/2<=2*(x/2)²=x²/2 鍥犱负 lim0=lim(x->0)x²/2=0 鎵浠 鐢卞す閫煎噯鍒欙紝寰 褰搙瓒嬭繎浜0鏃秎im1-cosx=0 鍗 褰搙瓒嬭繎浜0鏃秎imcosx=1
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绛旓細limx->0 (1-cosx)/(ax^2)=1 涓婁笅姹傚 lim x->0 sinx/(2ax)=1 1/2a=1 a=1/2
绛旓細鍗充袱鑰呮瘮鍊肩殑鏋侀檺鏄1锛屾樉鐒 x瓒嬩簬0鐨勬椂鍊欙紝lim(x鈫0)(1-cosx)/ (x^2/2)=lim(x鈫0)(1-cosx)'/ (x^2/2)'娲涘繀杈炬硶鍒欙紝鍚屾椂姹傚 =lim(x鈫0)sinx /x =1 鎵浠1-cosx鍜寈^2 /2鏄瓑浠风殑 鑰(1-cosx)/x鐨勬瀬闄愬煎氨涓嶆槸1浜嗭紝褰撶劧涓嶈兘闅忎究鏇挎崲 浣犺繖鏍锋兂锛岃鏄彧瑕佺瓑浜0浜嗗氨鑳芥浛鎹紝...
绛旓細棰樼洰鏋侀檺绗﹀悎缃楀繀濉旀硶鍒欙紝鍒欙細limx鈫0锛1-cosx锛/xsinx =limx鈫0锛坰inx锛/(sinx+xcosx)=limx鈫0锛坰inx/x锛/(sinx/x+cosx)=1/(1+1)=1/2
绛旓細鍙互鐢ㄦ礇蹇呰揪娉曞垯鍟婏紝浣嗙敤娲涘繀杈炬硶鍒欐眰瑙ｉ夯鐑︼紝鑷冲皯涓ゆ 鐢ㄧ瓑浠蜂唬鎹㈡瘮杈冪畝鍗 1-cosx绛変环浜1/2x^2 1-cos(鏍瑰彿x)绛変环浜1/2x lim(x瓒嬩簬0锛夛紙1-cosx)/x(1-cos(鏍瑰彿x)=lim(x瓒嬩簬0锛1/2x^2/(x*1/2x)=1
