等价无穷小的证明 高数中的等价无穷小要怎么证明
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x\u21920\u65f6\uff0clim\uff08sinx\uff0dx\uff09/Ax^k\uff1d\uff0d1/2Ak\u00d7lim x^(3\uff0dk)\u3002
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\u65b9\u6cd5\u4e8c\uff1a
sinx\u5728x\uff1d0\u5904\u7684Taylor\u516c\u5f0f\u662fsinx\uff1dx\uff0d1/3!\u00d7x^3\uff0b1/5!\u00d7x^5+...\uff0c\u6240\u4ee5sinx\uff0dx\uff1d\uff0d1/6\u00d7x^3\uff0d1/120\u00d7x^5+...\uff1d\uff0d1/6\u00d7x^3(1+1/20\u00d7x^2+...)\u3002
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x\u21920\u65f6\uff0csinx\uff0dx\uff5e\uff0d1/6\u00d7x^3\u3002
\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
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绛旓細瑙ｏ細璇佹槑锛=limx-0arcsinx=arcsin0=0 limx-0x=0 浜岃呴兘=鏄鏃犵┓灏忛噺銆俵imx-0 arcsinx/x 鎹㈠厓娉曪細浠=arcsinx sint=sinarcsinx=x x-0,t-arcsin0=0,t-0 limt-0 t/sint lmt-0 t=0 limt-0 sint=sin0=0 鍒嗗瓙鍒嗘瘝閮借秼鍚戝唴浜0 0/0鍨 娲涘繀杈炬硶鍒欍1/cost(t-0)=1/cos0=1/1=...
绛旓細浠=e^x-1锛屼袱杈瑰彇瀵规暟锛屽垯鏈墄=ln(y+1)lim(x鈫0)e^x-1 / x =lim(y鈫0)y / ln(y+1)=1 / lim(y鈫0)ln(y+1)/y =1 / lim(y鈫0)ln(y+1)^1/y =1 / 1 =1 璇佹槑锛歭im(y鈫0)ln(y+1)^1/y=e 绛変环鏃犵┓灏鏄棤绌峰皬涔嬮棿鐨勪竴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▼...
绛旓細绛変环鏃犵┓灏鍙互閫氳繃浠ヤ笅鏂瑰紡鎺ㄥ锛1銆佹瀬闄愮殑瀹氫箟锛氱瓑浠锋棤绌峰皬鏄熀浜庢瀬闄愮殑姒傚康鎺ㄥ鍑烘潵鐨勩傚湪涓瀹氱殑鏉′欢涓嬶紝褰撹嚜鍙橀噺x瓒嬭繎浜庢煇涓偣a鏃讹紝鍑芥暟f锛坸锛夌殑鍊艰秼杩戜簬涓涓父鏁癆锛屽垯绉癴锛坸锛夊湪x=a澶勬瀬闄愪负A銆傝岀瓑浠锋棤绌峰皬鍒欐槸閫氳繃灏嗘棤绌峰皬閲忚〃绀轰负鍏锋湁鐩稿悓鏋侀檺鐨勫彟涓涓棤绌峰皬閲忥紝浠庤屽疄鐜颁簡鐩镐簰鏇挎崲鐨勭洰鐨勩
绛旓細鍏蜂綋璇佹槑杩囩▼濡備笅锛歩m (1+1/x)^x =lim e^[ ln ((1+1/x)^x)]= e^ lim [ x ln (1+1/x)]x-->鏃犵┓澶 1/x--> 0 姝ゆ椂锛宭n (1+1/x) = 1/x 锛绛変环鏃犵┓灏锛塴im [ x ln (1+1/x)] = x * 1/x = 1 鍘熷紡= e^ 1 = e 鏁板垪鏋侀檺 璁 {Xn} 涓哄疄鏁板垪锛宎 涓哄畾鏁...
绛旓細1銆绛変环鏃犵┓灏忕殑鏁欏锛屽湪鍥藉鏄喎鍐锋竻娓咃紝鍥藉唴鏄哺娌告壃鎵傝冩煡鍒濆鑰咃紝鑻ヤ笉鑰冪瓑浠锋棤绌峰皬锛屾暀甯堝氨涓嶇煡閬撳嚭浠涔堥銆2銆佹湰棰鐨勮瘉鏄鍙互鐢ㄤ袱绉嶆柟娉曪細鏂规硶涓锛氫粠澶栧埌閲岋紝姹傛瘮鍊硷紝鍙栨瀬闄愩傜粨鏋滆嫢绛変簬涓锛屽氨鏄瓑浠凤紱缁撴灉鑻ョ瓑浜庝笉鏄竴鐨勫父鏁帮紝灏辨槸鍚屼环锛涚粨鏋滆嫢鏄0锛屽垯鍒嗗瓙灏辨槸楂樹环鏃犵┓灏忥紱缁撴灉鑻ユ槸鈭烇紝...
绛旓細璇佹槑濡備笅锛鏃犵┓灏忕殑鎬ц川鏄細1銆佹湁闄愪釜鏃犵┓灏忛噺涔嬪拰浠嶆槸鏃犵┓灏忛噺銆2銆佹湁闄愪釜鏃犵┓灏忛噺涔嬬Н浠嶆槸鏃犵┓灏忛噺銆3銆佹湁鐣屽嚱鏁颁笌鏃犵┓灏忛噺涔嬬Н涓烘棤绌峰皬閲忋4銆佺壒鍒湴锛屽父鏁板拰鏃犵┓灏忛噺鐨涔樼Н涔熶负鏃犵┓灏忛噺銆5銆佹亽涓嶄负闆剁殑鏃犵┓灏忛噺鐨勫掓暟涓烘棤绌峰ぇ锛屾棤绌峰ぇ鐨勫掓暟涓烘棤绌峰皬銆6銆佹棤绌峰皬閲忎笉鏄竴涓暟锛屽畠鏄竴涓...
绛旓細璇佹槑濡備笅锛歟^x~x lim锛坸鈫0锛(a^x-1)/xlna=lim锛坸鈫0锛(e^xlna-1)/xlna 璁総=xlna 褰搙鈫0锛宼鈫0 鎵浠ュ師寮=lim锛坱鈫0锛塭^t-1/t=t-1/t=1 鎵浠^x-1鐨勭瓑浠锋棤绌峰皬鏄痻lna 绛変环鏃犵┓灏忕殑鎰忎箟锛氱瓑浠锋棤绌峰皬涓鑸彧鑳藉湪涔橀櫎涓浛鎹紝鍦ㄥ姞鍑忎腑鏇挎崲鏈夋椂浼氬嚭閿欙紙鍔犲噺鏃跺彲浠ユ暣浣撲唬鎹紝涓嶄竴瀹...
绛旓細璇佹槑绛変环鏃犵┓灏忕殑涓変釜鎬ц川锛氭礇蹇呰揪娉曞垯锛孾ln(1+x)]'=1/(x+1) [e^x-1]'=e^x 鍒嗘瘝瀵兼暟閮芥槸1锛岄偅涓嶅氨鍒嗗埆鍙樻垚浜1/(1+x)鍜宔^x褰搙鈫0鏃剁殑鏋侀檺銆傛棤绌峰皬鐨勭瓑浠峰叧绯诲叿鏈変笅鍒楁ц川锛1锛夛紝伪~伪鐨勮嚜鍙嶆э紙2锛夛紝鑻ノ眫尾锛屽垯尾~伪锛堝绉版э級锛屽洜涓何辨槸鏃犵┓灏忎笖lim锛埼/伪锛=1锛屾墍浠ノ眫...
绛旓細涓銆佸父鐢绛変环鏃犵┓灏鏇挎崲鍏紡琛ㄥ強璇佹槑 褰搙瓒嬭繎浜0鏃:e^x-1~x銆乴n(x+1)~x銆乻inx~x銆乤rcsinx~x銆乼anx~x銆乤rctanx~x銆1-cosx~ (x^2)/2銆乼anx-sinx~(x^3)/2銆(1+bx)^a-1~abx銆備簩銆佹墿灞曠煡璇 1銆佹棤绌峰皬 鏃犵┓灏忛噺鏄暟瀛﹀垎鏋愪腑鐨勪竴涓蹇碉紝鍦ㄧ粡鍏哥殑寰Н鍒嗘垨鏁板鍒嗘瀽涓紝鏃犵┓灏忛噺閫氬父...
绛旓細涓夎鍑芥暟涓庡弽涓夎鍑芥暟鐨绛変环鏃犵┓灏 sinx ~ x锛氳繖涓熀鏈瀬闄愬湪銆婇珮绛夋暟瀛︺嬪悓娴庣増鐨勮瘉鏄涓湁鐫璇︾粏鐨勯槓杩帮紝鍏朵腑宸у鍦拌繍鐢ㄤ簡杈呭姪鍦嗗拰澶归煎噯鍒欍1-cosx ~ 1/2 x²锛氬叧閿湪浜庡埄鐢ㄤ笁瑙掑嚱鏁扮殑浜屽嶈鍏紡锛屽綋2x鏇挎崲涓簒鏃讹紝缁撳悎绛変环鏃犵┓灏弒inx~x锛屾垜浠緱鍒拌繖涓閲嶈缁撴灉銆傚叾浠栫瓑浠锋棤绌峰皬鎺ㄥ tanx ~ ...
