高数等价无穷小代换
书中应该漏掉了e∧，解析如图
绛旓細绛変环鏃犵┓灏鐨勫疄璐ㄥ氨鏄 x瓒嬩簬x0鏃讹紝f(x0)鍜実(x0)閮借秼浜0 鑰宖(x)/g(x) 瓒嬩簬1 鍗宠涓篺(x) ~g(x)绛変环鏃犵┓灏忎竴鑸彧鑳藉湪涔橀櫎涓鏇挎崲锛岃屽湪鍔犲噺涓浛鎹㈠父浼氬嚭閿 浣犺繖鏍锋潵鎯筹紝姹俧(x)/h(x)鐨勬瀬闄愬 鑰宖(x)鍜実(x)鏄瓑浠锋棤绌峰皬 閭ｄ箞f(x)/h(x)=f(x)/g(x) *g(x)/h(x)浠ｅ叆f(...
绛旓細鏉′欢锛1銆佽浠ｆ崲鐨勯噺锛屽湪鍙栨瀬闄愮殑鏃跺欐瀬闄愬间负0锛2銆佽浠ｆ崲鐨勯噺锛屼綔涓鸿涔樻垨鑰呰闄ょ殑鍏冪礌鏃跺彲浠ョ敤绛変环鏃犵┓灏忎唬鎹锛屼絾鏄綔涓哄姞鍑忕殑鍏冪礌鏃跺氨涓嶅彲浠ャ備簨瀹炰笂锛岀瓑浠锋棤绌峰皬鏄敱娉板嫆鍏紡鎺ㄥ鑰屾潵锛屾墍浠ヨ繍鐢ㄧ瓑浠锋棤绌峰皬鐨勭粨璁哄氨鏄紝涔橀櫎鍙互鏁翠綋鎹紝鑰屽姞鍑忔儏鍐典笉鑳芥崲锛屽嵆浣垮彲浠ワ紝閭ｄ篃鏄噾宸ф纭備笅闈㈢粰鍑轰粈涔...
绛旓細楂樻暟閲嶈鐨绛変环鏃犵┓灏忎唬鎹涔嬩竴锛氾紙1+x锛塣a -1 ~ ax 鈫掋愶紙1+x锛塣a ~ 1+ ax銆绛変环浠ｆ崲锛屾妸x鎻愬嚭鏉ワ紝閲岃竟鐨勫紡瀛愬氨鏄秼杩戜簬1鐨勪唬鏁板紡銆倀 瓒嬩簬0锛氣垰锛1+t锛=1+t/2 銆愶紙1+t锛塣1/2=1+t/2銆(1+t)^a=1+at+o(t)鍙互绛変环(1+t)^a ~ 1+at 杩欎釜濡傛灉a鏄暣鏁...
绛旓細姝ｇ‘鍋氭硶鏄厛閫氬垎锛屽啀鐢ㄦ礇蹇呰揪娉曞垯鎴朤aylor灞曞紡銆傜浜岄绫讳技锛(1+1/x)^x鐩稿鏁翠釜琛ㄨ揪寮忎笉鏄洜瀛愶紝鍥犳涓嶈兘绛変环鏇挎崲銆傛纭仛娉曟槸锛氬厛鍙栧鏁帮紝鐒跺悗鐢ㄦ礇蹇呰揪娉曞垯鎴朤aylor灞曞紡锛屽缓璁敤 Taylor灞曞紡銆傚彇瀵规暟鍚庯紝lim x*(1锛峹ln(1+1/x))锛漧im x*(1锛峹銆1/x锛1/(2x^2)+灏弌(1/x^2)銆)锛...
绛旓細楂樼瓑鏁板鍒╃敤绛変环鏃犵┓灏忎唬鎹姹傛瀬闄恖im,<x-0>(1-cos3x)/ln(1+x2) x->0 1-cos3x绛変环浜9xx/2 ln(1+xx)绛変环浜巟x 鎵浠ュ師寮弆imx->0(9xx/2)/xx=9/2 鍒╃敤绛変环鏃犵┓灏忔浛鎹 lim鐨剎瓒嬪悜浜0 閭n(1+x/2)鏄灏 x->0,ln(1+x/2)->0 x->0,sin(x/6)->0 (x/6) 鏄...
绛旓細鏄晩銆倄瓒嬩簬0鏃跺欙紝姹傛瀬闄愶紝鍙互杩愮敤绛変环鏃犵┓灏鏉ユ眰瑙ｃ倄瓒嬩簬0鏃跺欙紝姹俧锛坸²/sin²x锛変篃鍙互浣跨敤绛変环鏃犵┓灏忔眰瑙ｃ倄²鍜宻in²x鏄瓑浠锋棤绌峰皬锛屾墍浠ュ彲浠ユ眰寰楀嚱鏁扮殑鏋侀檺銆傜瓑浠锋棤绌峰皬锛楂樻暟涓父鐢ㄤ簬姹倄瓒嬩簬0鏃跺欐瀬闄愶紝褰撶劧锛寈瓒嬩簬鏃犵┓鐨勬椂鍊欎篃鍙眰锛岃浆鍖栨垚鍊掓暟鍗虫垚涓虹瓑浠锋棤绌峰皬銆
绛旓細鎴戜滑鍦ㄥ仛绛変环鏃犵┓灏忔浛鎹鐨勬椂鍊欎竴瀹氳娉ㄦ剰锛屽姞鍑忓師鍒欎笂涓嶆浛鎹㈢殑(鎷嗗垎鍚庢瀬闄愰兘瀛樺湪鍙互鏇挎崲)锛屼箻绉彲浠ユ浛鎹傚姞鍑忎笉鍙互鏇挎崲锛屾槸鍥犱负鏇挎崲鍚庡皢鍘熷紡涓殑楂橀樁鏃犵┓灏忛」蹇界暐浜嗐傛棰樺氨鏄繖鏍风殑锛宻inx=x-x^3/3!+O(x^3)锛屽鏋滀綘鐩存帴鏇挎崲鐨勮瘽锛岄偅涔堝氨绛変簬鎶妜^3椤瑰拷鐣ヤ簡锛岃屾椤规鏄喅瀹氭鏋侀檺鐨勯噸瑕侀」銆
绛旓細1銆绛変环鏃犵┓灏忎唬鎹 ln(1+1/x)~1/x 鍘熷紡=lim(x->鈭) (3x^3+5)/(5x^3+3x)=3/5 2銆佸師寮=lim(x->鈭) {[1+3a/(x-a)]^[(x-a)/3a]}^3a*[1+3a/(x-a)]^a =e^3a =8 a=ln2 3銆佺瓑浠锋棤绌峰皬浠ｆ崲 e^x-1~x 1-cosx~(1/2)x^2 sinx~x 鍘熷紡=lim(x->0) ...
绛旓細璇存槑锛氬彲浠ヨ繖鏍绛変环浠ｆ崲銆傝嫢浣犱笉纭畾绛変环浠ｆ崲鏄惁姝ｇ‘锛屾棰樺彲浠ョ敤閲嶈鏋侀檺娉曟眰瑙ｃ傝В锛氾紙閲嶈鏋侀檺娉曪級鍘熷紡=lim(t->0){[(1+t²)^(1/t²)]^(t²cost²/sint²)} ={lim(t->0)[(1+t²)^(1/t²)]}^[lim(t->0)(t²cost²/sint&...
绛旓細姝ら鏄楂樼瓑鏁板涓鍏充簬绛変环鏃犵┓灏鐨勯鐩紝鍏抽敭鐐瑰氨鏄瓑浠锋棤绌峰皬鐨浠ｆ崲銆傜瓑浠锋棤绌峰皬閲屾湁杩欐牱涓涓叕寮忥細褰搙->0鏃(1+x)^a-1绛変环浜巃x 鎵浠ュ湪杩欓噷濂楃敤姝ゅ叕寮忋2娆℃牴鍙峰氨鐩稿綋浜0.5娆℃柟鍢涖(1+ax^2)^0.5-1绛変环浜1/2ax^2锛涘彟澶栦竴鏂归潰(sinx)^2绛変环浜巟^2杩欎釜鎴戜笉鐢ㄥ瑙ｉ噴浜嗗惂銆傚凡鐭ヤ袱鑰呬负绛変环...
