等价无穷小的证明 如何证明两函数为等价无穷小量？
\u9ad8\u7b49\u6570\u5b66\u7b49\u4ef7\u65e0\u7a77\u5c0f\u66ff\u6362\u8bc1\u660e\uff0c\u8c01\u80fd\u7ed9\u6211\u8bc1\u660e\u4e00\u4e0b\uff08\u8981\u8fc7\u7a0b\uff09\uff1f\u6d1b\u5fc5\u8fbe\u6cd5\u5219\uff0c[ln(1+x)]'=1/(x+1) [e^x-1]'=e^x \u5206\u6bcd\u5bfc\u6570\u90fd\u662f1\uff0c\u90a3\u4e0d\u5c31\u5206\u522b\u53d8\u6210\u4e861/(1+x)\u548ce^x\u5f53x\u21920\u65f6\u7684\u6781\u9650\u3002
lim(x->0) ( 1- cosx) /(x^2/2)
=lim(x->0) 2( 1- cosx) / x^2 (0/0 \u5206\u5b50\u5206\u6bcd\u5206\u522b\u6c42\u5bfc)
=lim(x->0) 2sinx/(2x)
=1
1- cosx ~ x^2/2
\u65e0\u7a77\u5c0f\u7684\u6027\u8d28\uff1a
1\u3001\u6709\u9650\u4e2a\u65e0\u7a77\u5c0f\u91cf\u4e4b\u548c\u4ecd\u662f\u65e0\u7a77\u5c0f\u91cf\u3002
2\u3001\u6709\u9650\u4e2a\u65e0\u7a77\u5c0f\u91cf\u4e4b\u79ef\u4ecd\u662f\u65e0\u7a77\u5c0f\u91cf\u3002
3\u3001\u6709\u754c\u51fd\u6570\u4e0e\u65e0\u7a77\u5c0f\u91cf\u4e4b\u79ef\u4e3a\u65e0\u7a77\u5c0f\u91cf\u3002
4\u3001\u7279\u522b\u5730\uff0c\u5e38\u6570\u548c\u65e0\u7a77\u5c0f\u91cf\u7684\u4e58\u79ef\u4e5f\u4e3a\u65e0\u7a77\u5c0f\u91cf\u3002
5\u3001\u6052\u4e0d\u4e3a\u96f6\u7684\u65e0\u7a77\u5c0f\u91cf\u7684\u5012\u6570\u4e3a\u65e0\u7a77\u5927\uff0c\u65e0\u7a77\u5927\u7684\u5012\u6570\u4e3a\u65e0\u7a77\u5c0f\u3002
\u9996\u5148\uff0c\u4e24\u4e2a\u51fd\u6570\u5fc5\u987b\u662f\u65e0\u7a77\u5c0f\uff0c\u5176\u6b21\u4e24\u4e2a\u51fd\u6570\u76f8\u9664\u5728\u540c\u4e00\u4e2a\u6570\u91cf\u7ea7\uff08\u5c31\u662fx^a\u6b21\u65b9\uff09\u4e0a\u662f\u7b49\u4e8e1.
可以直接相除求极限，根据某定理再分号上下求导值不变，上下求导得1/(1+x^2)，极限为1，所以等价绛旓細绛旓細鐢ㄤ簩鍊嶈鍏紡锛歝os2a=1-2sin²a 1-cos2a=2sin²a 鎵浠ワ細1-cosx=2sin²(x/2)~2脳(x/2)²~x²/2 鎵浠ワ細1-cosx鐨绛変环鏃犵┓灏涓簒²/2 绛変环鏃犵┓灏忔槸鏃犵┓灏忎箣闂寸殑涓绉嶅叧绯伙紝鎸囩殑鏄細鍦ㄥ悓涓鑷彉閲忕殑瓒嬪悜杩囩▼涓紝鑻ヤ袱涓棤绌峰皬涔嬫瘮鐨勬瀬闄愪负1锛屽垯绉拌繖涓...
绛旓細鐔熻甯哥敤绛変环鏃犵┓灏閲忓強鍏跺拰宸備竴鑸儏褰紝浣跨敤娲涘繀杈撅紙L\\'Hospital锛夋硶鍒欙紝鎴栬匱aylor鍏紡銆備妇渚嬶細x鈫0鏃讹紝sinx锛峹鐨勭瓑浠锋棤绌峰皬閲忥紵鏂规硶涓锛氳x鈫0鏃讹紝sinx锛峹锝濧x^k銆侫锛宬寰呭畾銆傜敱娲涘繀杈炬硶鍒欙紝x鈫0鏃讹紝lim锛坰inx锛峹锛/Ax^k锛漧im锛坈osx锛1锛/Akx^(k锛1)锛屽垎瀛愭浛鎹负绛変环鏃犵┓灏忛噺锛1/2...
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绛旓細lim(x->0) ( 1- cosx) /(x^2/2)=lim(x->0) 2( 1- cosx) / x^2 (0/0 鍒嗗瓙鍒嗘瘝鍒嗗埆姹傚)=lim(x->0) 2sinx/(2x)=1 => 1- cosx ~ x^2/2
