幂指函数求极限公式证明 幂指函数求极限的方法有哪些？
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绛旓細鏈変袱绉嶆柟娉曘涓绉嶅氨鏄埄鐢╢(x)=h(x)^g(x)=e^ln[h(x)^g(x)]=e^[g(x)lnh(x)]锛屽彟涓绉嶅埄鐢ㄤ袱杈瑰彇瀵规暟锛屽嵆lnf(x)=g(x)lnh(x)銆傚畠浠疄璐ㄦ槸涓浼氫簨
绛旓細1. 灏嗗箓鎸囧嚱鏁拌〃绀轰负鑷劧鎸囨暟鍑芥暟鐨勫舰寮忥紝鍗砤^x = exp(x * ln(a))銆2. 褰搙瓒嬭繎浜庢棤绌峰ぇ鏃讹紝x * ln(a)涔熻秼杩戜簬鏃犵┓澶с3. 鏍规嵁鑷劧鎸囨暟鍑芥暟鐨勬ц川锛屽綋鑷彉閲忚秼杩戜簬鏃犵┓澶ф椂锛宔xp(x)涔熻秼杩戜簬鏃犵┓澶с傚洜姝わ紝骞傛寚鍑芥暟a^x鍦▁瓒嬭繎浜庢棤绌峰ぇ鏃剁殑鏋侀檺鍙互琛ㄧず涓轰互e涓哄簳鐨勬寚鏁板嚱鏁癳xp(x * ln...
绛旓細鍥犱负鈥滃箓鎸囧瀷鈥鍑芥暟鏋侀檺姹傝В鏈鏅亶銆佹渶涓鑸殑鏂规硶锛屽埄鐢ㄧ殑鏄箓鎸囧瀷閫氳繃鍙栧鏁板彲浠ヨ浆鍖栦负澶嶅悎鍑芥暟鐨鐗圭偣銆傜敱浜巐nf(x)g(x)=g(x)lnf(x)锛宖(x)g(x)=eg(x)lnf(x)銆傚鍥炬墍绀猴細浣滀负骞傚嚱鏁锛屽叾骞傛寚鏁扮‘瀹氫笉鍙橈紝鑰屽箓搴曟暟涓鸿嚜鍙橀噺锛涚浉鍙嶅湴锛屾寚鏁板嚱鏁板嵈鏄簳鏁扮‘瀹氫笉鍙橈紝鑰屾寚鏁颁负鑷彉閲忋骞傛寚鍑芥暟灏辨槸...
绛旓細缁撹浜旇鏄庢棤绌峰皬鎵鍦ㄧ殑搴曞嚱鏁伴儴鍒唂(x)涓庢棤绌峰ぇ鎵鍦ㄧ殑鎸囧嚱鏁伴儴鍒唃(x)鐩镐箻锛屽叾涔樼Н鏋侀檺濡備负a锛屽垯1^鈭炲瀷鐨勫箓鎸囧嚱鏁癧1+f(x)]^[g(x)]鐨勬瀬闄愪负e^a銆傜粨璁哄叚銆佺粨璁轰竷銆佺粨璁哄叓锛氭敞鎰忕粨璁轰竷宸︾鏋侀檺鐨勮繃绋嬒(x)->鈭烇紝鑰屼笖鍒嗗瓙銆佸垎姣嶅強鎸囨暟鐨勫彉閲忓畬鍏ㄧ浉鍚岋紝鍧囦负蠁(x)銆傜粨璁轰竷宸︾鐨骞傛寚鍑芥暟鐨勬瀬...
绛旓細y=f^g lny=glnf limlny=limglnf=limg limlnf =limg lnlimf =BlnA limy=A^B
绛旓細骞傛寚鍑芥暟姹傛瀬闄鏂规硶褰掔撼濡備笅锛氭柟娉曚竴锛氶兘鏄箓鎸囨暟鐨勫舰寮忥紝鍙互鎻愬嚭鏈楂樻椤癸紝鏋侀檺鍊煎氨鏄渶楂樻椤圭殑绯绘暟涔嬫瘮锛屽涓嬪浘鎵绀恒傛柟娉曚簩锛氬彲浠ョ敤娲涘繀杈炬硶鍒欐眰鏋侀檺銆傚叿浣撳仛娉曟槸鍚屾椂瀵瑰垎瀛愬垎姣嶆眰瀵硷紝鐒跺悗鍊熷姪鏂规硶涓鎴栬呯洿鎺ヤ唬鍏ワ紝鍙互寰楀埌绛旀銆傚悓搴曟暟骞傜殑闄ゆ硶鏄暣寮忛櫎娉曠殑鍩虹锛岃鐔熺粌鎺屾彙銆傚悓搴曟暟骞傜殑闄ゆ硶娉曞垯鏄...
绛旓細1銆佲滅‘瀹氬瀷鈥濓紝鑻鈫抋锛寁鈫抌锛屽叾涓細a>0锛屽垯lim u^v=a^b 2銆佲滀笉瀹氬瀷鈥濓紝鍖呮嫭0^0锛屸垶^0锛1^鈭炵瓑锛岃繖浜涗竴鑸彲鍋氫互涓嬩袱绉嶅彉鍖 u^v=e^(vlnu)鎴栧彇瀵规暟锛寉=u^v锛屽垯lny=vlnu锛岃繖鏍峰彲浠ユ妸骞傛寚鍑芥暟鐨勬瀬闄闂杞寲涓哄叾瀹冪被鍨嬶紱3銆佸浜1^鈭炲瀷锛岃繕鏈夊彟涓绉嶆柟娉曪紝灏辨槸鍒╃敤绗簩涓噸瑕佹瀬闄愩...
绛旓細瑙ｏ細lim(x->0)[(e^x+x)^(1/x)]=lim(x->0){e^[ln(e^x+x)/x]} (搴旂敤瀵规暟鎬ц川鍙栧鏁)=e^{lim(x->0)[ln(e^x+x)/x]} (搴旂敤鍒濈瓑鍑芥暟鐨杩炵画鎬)=e^{lim(x->0)[(e^x+1)/(e^x+x)]} (0/0鍨鏋侀檺锛屽簲鐢ㄧ綏姣旇揪娉曞垯)=e^[(1+1)/(1+0)]=e^2 lim(x->0){...
绛旓細鍒╃敤x=e^lnx,鐒跺悗鍒╃敤娲涘繀杈炬硶鍒欒繘琛姹傛瀬闄锛屽氨鍙互寰堝揩寰楀嚭缁撴灉涓篹^(-1/6).
绛旓細1. 甯告暟鍑芥暟鏋侀檺鍏紡锛歭im(xa) c = c锛屽叾涓璫鏄竴涓父鏁般傝繖鎰忓懗鐫褰撹嚜鍙橀噺x瓒嬩簬鏌愪釜鍊糰鏃讹紝甯告暟鍑芥暟鐨勬瀬闄愬间负璇ュ父鏁癱銆2. 骞傚嚱鏁版瀬闄愬叕寮锛歭im(xa) x^n = a^n锛屽叾涓璶涓烘鏁存暟銆傚綋鑷彉閲弜瓒嬩簬鏌愪釜鍊糰鏃讹紝骞傚嚱鏁扮殑鏋侀檺鍊肩瓑浜庤鍊糰鐨刵娆″箓銆3. 鎸囨暟鍑芥暟鏋侀檺鍏紡锛歭im(xa) a^x = a^...
