lim(x→0)(1/cosx)=?
1/cosx在x=0处连续,直接代值即可lim(x→0)(1/cosx)=1/cos0=1
绛旓細鎮ㄧ殑杈撳叆鏈夎銆傚彲鑳芥槸鍏充簬閲嶈鏋侀檺鍏紡鐨勯棶棰樸傝鎯呭鍥炬墍绀:渚涘弬鑰冿紝璇风瑧绾炽
绛旓細e鐨1/x娆℃柟锛屽綋x瓒嬭繎0鏃讹紝瀹冪殑鏋侀檺涓嶅瓨鍦ㄣ傚洜涓哄乏鍙虫瀬闄愪笉鐩哥瓑銆傝В锛氬洜涓lim(x鈫0-)(1/x)=-鈭烇紝鑰宭im(x鈫0+)(1/x)=+鈭炪傞偅涔坙im(x鈫0-)e^(1/x)=e^(-鈭)=0銆俵im(x鈫0+)e^(1/x)=e^(+鈭)=+鈭炪傚垯lim(x鈫0-)e^(1/x)鈮爈im(x鈫0+)e^(1/x)銆傚嵆lim(x鈫0)(1...
绛旓細鍘熷紡=lim(x鈫0)(1-2x)^[(-1/2x)*(-2x)*(1/x)]=e^[lim(x鈫0)(-2x)*(1/x)]=e^(-2)=1/e²銆傛牴鎹叕寮弆im(x鈫0)(1+x)^(1/x)=e 渚嬪锛氬師寮=limx鈫0(1+x/2)*[(1+x/2)^2/x]^(-1/2)=(1+0)*e^(-1/2)=e^(-1/2)銆(1-1/x)^2x ={[1+(-1...
绛旓細lim(x鈫0) ln(1+x)/x=lim(x鈫0) ln(1+x)^(1/x)=ln[lim(x鈫0) (1+x)^(1/x)]鐢变袱涓噸瑕佹瀬闄愮煡:lim(x鈫0) (1+x)^(1/x)=e,鎵浠ュ師寮=lne=1,鎵浠n(1+x)鍜寈鏄瓑浠锋棤绌峰皬 绛変环鏃犵┓灏忔槸鏃犵┓灏忕殑涓绉嶃傚湪鍚屼竴鐐逛笂锛岃繖涓や釜鏃犵┓灏忎箣姣旂殑鏋侀檺涓1锛岀О杩欎袱涓棤绌峰皬鏄瓑浠...
绛旓細褰x瓒嬪悜浜0鏃讹紝ln|x|瓒嬪悜浜庤礋鏃犵┓澶э紝鎵浠1/ln|x|瓒嬪悜浜0
绛旓細lim(x鈫0)(1+x)^lnx =(1+x)^(1/x)*(xlnx)=e^(xlnx)姹倄lnx鐨勬瀬闄 杞崲鎴 =lnx/(1/x)娲涘繀杈炬硶鍒欏垎瀛愬垎姣嶄笂涓嬫眰瀵 =1/x /(-1/x²)= -x ~0 鎵浠ュ師寮 =e^0 =1 甯屾湜鑳借В鍐充綘鐨勭枒闂甇鈭鈭㎡~
绛旓細1銆侀噸瑕佹瀬闄愶紝涓嶅湪浜庢槸瓒嬪悜浜庢棤绌峰ぇ锛岃繕鏄秼鍚戜簬0锛屾垨瓒嬪悜浜庝换浣曚竴涓父鏁帮紝鑰屽湪浜庡畠鐨勬瀬闄愬舰寮忋備篃灏辨槸锛屾嫭鍙峰唴锛屽繀椤绘槸 1 + 鏃犵┓灏忥紝鎷彿澶栧繀椤绘槸 鏃犵┓澶ф骞傘傛嫭鍙峰唴鐨勬棤绌峰皬璺熸嫭鍙峰鐨勬棤绌峰ぇ娆″箓蹇呴』鏄掓暟鍏崇郴銆2銆佸叿浣撶ず渚嬪涓嬶紝甯屾湜鑳介氳繃杩20涓緥瀛愶紝妤间富鎮熷嚭瀹冧滑鐨勫疄璐ㄦ剰鎬濄
绛旓細棣栧厛闇瑕佽y=(1+1/x)^x锛屼袱杈瑰悓鏃跺彇鑷劧瀵规暟寰 lny=xln(1+1/x)=[ln(1+1/x)]/(1/x)鐢辨礇蹇呰揪娉曞垯lny=lim銆恱鈫掆垶銆慬ln(1+1/x)]/(1/x)=[1/(1+1/x)] (1/x) '/(1/x)'=1/(1+1/x)=1 鎵浠=e銆恱鈫掆垶銆 鍗lim(x鈫鈭) (1+1/x)^x=e銆
绛旓細lim(x鈫0)(x/sinx)=lim(x鈫0)(1/cosx) 锛堟礇蹇呰揪娉曞垯锛屽垎瀛愬垎姣嶅悓鏃舵眰瀵硷級=1/cos0 =1/1 =1 鍗砽im(x鈫0)x/sinx=1銆傚嵆lim(x鈫0)(x/sinx)绛変簬1銆傛礇蹇呰揪娉曞垯鏄湪涓瀹氭潯浠朵笅閫氳繃鍒嗗瓙鍒嗘瘝鍒嗗埆姹傚鍐嶆眰鏋侀檺鏉ョ‘瀹氭湭瀹氬紡鍊肩殑鏂规硶銆傞浂姣旈浂鍨 鑻ュ嚱鏁癴(x)鍜実(x)婊¤冻涓嬪垪鏉′欢lim(x鈫...
绛旓細璇佹槑锛氬浜庝换鎰忕殑蔚>0, 鍙栁1=蔚*sqrt(a)>0 ,鏍规嵁鏁板垪鏋侀檺鐨勫畾涔夛紝鏈夛細瀵逛簬蔚1锛屽瓨鍦∟灞炰簬N+锛屽綋n>N鏃讹紝鏈墊Xn-a|<蔚1鎴愮珛 (1寮忋傜敱浜嶺n>0,a>0, 鏈夛細|sqrt(Xn)-sqrt(a)|=|Xn-a|/(sqrt(Xn)+sqrt(a))銆備笖鐢变簬sqrt(Xn)>0,鍒 sqrt(Xn)+sqrt(a)>sqrt(a)锛屼簬鏄倈sqrt...
