求极限。第一题。x趋于无穷。
x\u8d8b\u4e8e\u65e0\u7a77\u65f6\uff0c(x/(x+1))^x\u6c42\u6781\u9650\u539f\u5f0f=lim(x->\u221e) [1-1/(x+1)]^x
=lim(x->\u221e) {[1+1/(-x-1)]^(-x-1)}^(-1)*[1+1/(-x-1)]^(-1)
=1/e
\u4f60\u597d\uff0c\u5982\u56fe\uff1a\u5e0c\u671b\u5e2e\u5f97\u5230\u4f60\uff0c\u6b63\u786e\u54e6\u671b\u91c7\u7eb3O(\u2229_\u2229)O\u54c8\u54c8~
∴lim(x→∞)[1-1/(2x)]^x=im(x→∞){[1+1/(-2x)]^(-2x)}^(-1/2)=e^(-1/2)。供参考。
绛旓細瑙ｏ細鍒╃敤鍩烘湰鏋侀檺鍏紡锛宭im(x鈫掆垶)(1+1/x)^x=e锛屽皢鈥-2x鈥濊浣滀竴涓暣浣擄紝鈭磍im(x鈫掆垶)[1-1/(2x)]^x=im(x鈫掆垶){[1+1/(-2x)]^(-2x)}^(-1/2)=e^(-1/2)銆備緵鍙傝冦
绛旓細lim(1-1/X)X锛X瓒嬩簬鏃犵┓锛姹傛瀬闄锛岃缁嗚繃绋嬶紝璋㈣阿 楂樹腑鏁板瑙ｆ硶 lim(x鈫掆垵) (1-1/X)^X =lim(x鈫掆垵) (1+1/(-X))^X =lim(x鈫掆垵) 1/[(1+1/(-X))^(-X)] =1/[lim(x鈫掆垵) (1+1/(-X))^(-X)] =1/e 楂樼瓑鏁板瑙ｆ硶 lim(x鈫掆垵) (1-1/X)^X =(浠...
绛旓細鍑℃槸姹傛瀬闄锛岃秼鍚戜笌鏃犵┓澶ф椂锛屼笂鏉ュ氨鐪嬪垎瀛愬垎姣嶇殑娆★紝鍙湅楂樻骞傦紝鏈楂樻骞傚湪鍒嗗瓙灏辨槸鏃犵┓澶э紙涓嶅瓨鍦級锛屾渶楂樻骞傚湪鍒嗘瘝灏辨槸0锛屽鏋滃垎瀛愬垎姣嶄竴鏍凤紝灏辩瓑浜庢槸浠栦滑鍓嶉潰鐨勭郴鏁般x瓒嬪悜0鐪嬫渶浣庢骞傘
绛旓細瀵逛簬绗簩棰橈紝浣犲彧闇瑕佸鏋侀檺鏈夌畝鍗曠殑鐞嗚В锛鏃犵┓澶х殑鍜岋紝鍙敤鏈楂橀」銆傚嵆浣縳鏄棤绌凤紝瀵箈^2鑰岃█锛屼篃鏄彲浠ュ拷鐣ョ殑銆傛墍浠^2+x=x^2 鑰屽瓒嬩簬闆讹紝姹傚拰鍙湅鏈浣庢锛x瓒嬩簬闆舵椂锛岀浉瀵逛簬x鑰岃█锛寈^2鍙拷鐣ャ傚嵆x^2+x=x 鎵浠ユ嘲鍕掑叕寮廵^x灞曞紑椤逛腑绗笁椤瑰紑濮嬮氬父閮芥槸琚拷鐣ョ殑銆備篃灏辨槸ln(1+x)=x 鏃...
绛旓細=e^锛-2x/x+4锛=e^-2 =1/e^2 (X瓒嬩簬鏃犵┓)鈥鏋侀檺鈥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▼涓紝閫愭笎鍚戞煇涓涓‘瀹氱殑鏁板糀涓嶆柇鍦伴艰繎鑰屸滄案杩...
绛旓細绗簩绉嶆柟娉曟湁閿欒锛岄噸瑕鏋侀檺鐢ㄩ敊浜嗐侺=lim(x->+鈭) x^(1/x)lnL =lim(x->+鈭) lnx/x (鈭/鈭)=lim(x->+鈭) 1/x =0 L =e^0 =1 L=lim(x->+鈭) x^(1/x)=1
绛旓細棣栧厛锛屾牴鎹鏋侀檺鐨勫畾涔夛紝褰x瓒嬩簬0鐨勬椂鍊欙紝鎴戜滑鍙互灏嗗師寮忚浆鍖栦负浠ヤ笅褰㈠紡:lim(x鈫0⁺)[(arcsin(x))^x - x^x]/x^2ln^2(1+x)鎺ョ潃锛屾垜浠彲浠ヤ娇鐢ㄦ嘲鍕掑叕寮忓皢arcsin(x)鍜寈^x鍦▁=0澶勫睍寮:arcsin(x) = x - (1/6)x^3 + O(x^5)x^x = 1 + xln(x) + O(x^2)灏嗕笂杩板睍寮...
绛旓細鍙互璇存病鏈夊叧绯伙紝鍥犱负杩欎釜鐙珛姹傚緱鍑鏋侀檺鏉ワ紝褰x瓒嬩簬鏃犵┓鏃,1/x瓒嬩簬0锛岃宻in鏄繛缁嚱鏁帮紝鎵浠ョ洿鎺ヤ唬鍏0鍙互姹傚緱鏋侀檺鏄0銆備篃鍙互璇存湁涓鐐瑰叧绯伙紝瀹冩槸绗竴涓噸瑕佹瀬闄愮殑涓閮ㄥ垎銆
绛旓細娉板嫆鍏紡鏄皢涓涓湪x=x0澶勫叿鏈塶闃跺鏁扮殑鍑芥暟f(x)鍒╃敤鍏充簬(x-x0)鐨刵娆″椤瑰紡鏉ラ艰繎鍑芥暟鐨勬柟娉曘傛牴鎹甽n(1+x)=x-x^2/2 寰楀嚭ln(1+1/x)=1/x-1/x^2/2 寰楀嚭鏋侀檺=x-[x-1/2]=1/2 N鐨勭浉搴旀 涓鑸潵璇达紝N闅徫电殑鍙樺皬鑰屽彉澶э紝鍥犳甯告妸N鍐欎綔N(蔚)锛屼互寮鸿皟N瀵刮电殑鍙樺寲鑰屽彉鍖栫殑渚濊禆...
绛旓細瑙ｆ瀽锛氫笂涓嬪悓鏃堕櫎浠,寰楀師寮忥紳 lim(x鈫掆垶)(1锛2sinx/x)(3锛4cosx/x)锛(1锛0)/(3锛0)锛1/3.锛堝洜涓簒鈫掆垶,鎵浠1/x鈫0,鑰宻inx鍜宑osx涓烘湁鐣屽嚱鏁,鎵浠inx/x涓巆osx/x鍧囦负鏃犵┓灏,鍗斥啋0锛,6,sinx鍜宑osx姘歌繙鍦╗-1锛1]涔嬮棿锛屽綋x鏃犵┓澶ф椂锛屽拷鐣ヤ笉璁 鍘熷紡=x/3x=1/3 (1涓湴鐞+2涓...
