当x趋近于1时，1/lnx的极限是多少 当x趋近于0时，x＋1的极限是多少？
\u6781\u9650x\u8d8b\u8fd1\u4e8e1\u65f6\uff0c1/lnX\u7684\u6781\u9650lnx\u7684\u5bfc\u6570\u4e3a1/x\u6240\u4ee5\uff0c\u7b54\u6848\u662f1\uff0c\u671b\u91c7\u7eb3
\u672c\u9898\u89e3\u7b54\uff1a
\u5de6\u6781\u9650
=
-\u221e
\u53f3\u6781\u9650
=
+\u221e
\u56e0\u4e3a\uff0c\u5de6\u6781\u9650
\u2260
\u53f3\u6781\u9650\uff0c
\u6240\u4ee5\uff0c\u672c\u9898\u5728x=0\u5904\u7684\u6781\u9650\u4e0d\u5b58\u5728\u3002
\u8bf4\u660e\uff1a
1\u3001\u5982\u679c\u6781\u9650\u5b58\u5728\uff0c\u5fc5\u987b\u5de6\u3001\u53f3\u6781\u9650\u5b58\u5728\uff0c\u5e76\u4e14\u76f8\u7b49\u3002
\u4e5f\u5c31\u662f\uff1a\u53ea\u8981\u5de6\u6781\u9650\u4e0d\u5b58\u5728\uff0c\u6781\u9650\u5c31\u4e0d\u5b58\u5728\uff1b
\u53ea\u8981\u53f3\u6781\u9650\u4e0d\u5b58\u5728\uff0c\u6781\u9650\u5c31\u4e0d\u5b58\u5728\uff1b
\u53ea\u8981\u5de6\u6781\u9650\u3001\u53f3\u6781\u9650\u4e0d\u76f8\u7b49\uff0c\u6781\u9650\u5c31\u4e0d\u5b58\u5728\u3002
\u65e0\u8bba\u662f\u5de6\u6781\u9650\uff0c\u8fd8\u662f\u53f3\u6781\u9650\uff0c\u53ea\u8981\u51fa\u73b0\u65e0\u7a77\u5927\uff0c\u6781\u9650\u5c31\u4e0d\u5b58\u5728\uff01
2\u3001\u5982\u679c\u5f53x\u8d8b\u5411\u4e8e2\u65f6\uff0c\u5de6\u6781\u9650\u7b49\u4e8e3\uff0c\u53f3\u6781\u9650\u7b49\u4e8e4\u3002
\u6211\u4eec\u53ea\u8bf4\u5de6\u6781\u9650\u5b58\u5728\uff0c\u53ea\u8bf4\u53f3\u6781\u9650\u5b58\u5728\u3002\u6211\u4eec\u53ea\u8bf4\u5728x=2\u8fd9\u4e00\u70b9\u6781\u9650\u4e0d\u5b58\u5728\uff01
\u65e0\u8bba\u662f\u5de6\u6781\u9650\uff0c\u8fd8\u662f\u53f3\u6781\u9650\uff0c\u5982\u679c\u6211\u4eec\u8bf4\u5b83\u4e0d\u5b58\u5728\uff0c\u662f\u6307\uff1a
a\u3001\u4e0d\u8d8b\u5411\u4e8e\u4e00\u4e2a\u56fa\u5b9a\u503c\uff0c\u6216\u5927\u6216\u5c0f\uff0c\u6ca1\u6709\u56fa\u5b9a\u7684\u8d8b\u5411\u6027(tendency)\uff1b
b\u3001\u6709\u56fa\u5b9a\u7684\u8d8b\u5411\u6027\uff0c\u4f46\u4e0d\u662f\u56fa\u5b9a\u503c\uff0c\u800c\u662f\u8d8a\u6765\u8d8a\u5927\uff0c\u8d8b\u5411\u4e8e\u65e0\u7a77\u5927\u3002
3\u3001\u5728\u8d8b\u5411\u4e8e\u65e0\u7a77\u5927\u65f6\uff0c\u56e0\u4e3a\u5b83\u4e0d\u662f\u4e00\u4e2a\u5177\u4f53\u7684\u5f88\u5927\u7684\u6570\uff0c\u800c\u662f\u4e00\u4e2a\u8d8a\u6765\u8d8a\u5927
\u7684\u8fc7\u7a0b\uff0c\u7406\u8bba\u4e0a\u662f\u4e0d\u5b58\u5728\u3002\u4e0d\u8fc7\u4e3a\u4e86\u7528\u6570\u5b66\u7b26\u53f7\u628a\u8fd9\u4e00\u610f\u601d\u5b8c\u7f8e\u5730\u8868\u8fbe\u51fa\u6765\uff0c
\u56fd\u5185\u56fd\u5916\uff0c\u90fd\u91c7\u53d6\u4e86\u5171\u540c\u7684\u8bb0\u6cd5\uff1a
lim
1/x²
=
\u221e
\u8fd9\u53ea\u662f\u4e00\u4e2a\u628a\u6781\u9650\u662f\u6709\u9650\u503c\u4e0e\u65e0\u9650\u503c\u8054\u5408\u5728\u4e00\u8d77\u7684\u65b9\u6cd5\uff0c
x\u21920
\u4f46\u662f\uff0c\u8fd9\u79cd\u8bb0\u6cd5\uff0c\u5e76\u4e0d\u8868\u793a\u221e\u662f\u4e00\u4e2a\u5177\u4f53\u7684\u6570\u3002
4\u3001\u82f1\u8bed\u4e2d\uff0c\u4e0d\u5b58\u5728\u7684\u5199\u6cd5\u662f\uff1adne\uff0c\u6216
d.n.e.
=
do
not
exist.
\u5982\u679c\u697c\u4e3b\u8fd8\u6709\u7591\u95ee\uff0c\u8bf7hi\u6211\u3002
x→1-时1/lnx→-∞。
lim(x→1)[ 1/(1-x) -1/lnx]
=lim(x→1)[ lnx-(1-x)]/[(1-x)lnx]
=lim(x→1) [lnx+x-1]' / [lnx-xlnx]'
=lim(x→1)[1/x+1]/[1/x-lnx-1]
=∞
绛旓細褰搙鍚1鐨勫乏鏂瑰悜瓒嬭繎浜1鏃讹紝鍗硏灏忎簬1鏃讹紝鏋侀檺瓒嬪悜浜庤礋鏃犵┓澶э紝褰撳悜鍙虫柟鍚戣秼杩戜簬1鏃讹紝鏋侀檺瓒嬭繎浜庢鏃犵┓澶с倄/(x-1)=lim(x鈫1)1/(x-1)=鈭炲洜涓簂im(x鈫1)(x-1)=0涔熷氨鏄垎姣嶈秼鍚戜簬鏃犵┓灏忥紝鍊掕繃鏉ョ殑缁撴灉褰撶劧鏄棤绌峰ぇ銆傛牴鎹珮绛夋暟瀛︽瀬闄愬畾涔夛細鍑芥暟鏋侀檺涓烘棤绌峰ぇ鏃讹紝璁や负鏋侀檺涓嶅瓨鍦,杩欓噷鏆傛椂琛ㄨ堪...
绛旓細锛屾墍浠ュ乏鏋侀檺涓2鍊嶇殑e鐨勮礋鏃犵┓澶ф鏂圭瓑浜0;(X-1) X瓒嬪悜浜1鏃锛屽乏鏋侀檺鏄礋鏃犵┓澶э紝鍘熷紡鍖栫畝涓(X+1)e锝1锛忥紙X-1锛夛紝鍙虫瀬闄愭槸姝ｆ棤绌峰ぇ
绛旓細x鈫1鏃讹紝鏃犺鏄乏鏋侀檺锛岃繕鏄彸鏋侀檺锛岄兘鏄瓑浜1锛屾墍浠ヨ繖涓嚱鏁板湪x鈫1鏃剁殑鏋侀檺灏辨槸1锛屽拰鍑芥暟鍦▁=1鏃剁殑鍑芥暟鍊兼棤鍏炽
绛旓細褰搙瓒嬩簬0+鎴栬秼浜0-鏃讹紝鏋侀檺閮借秼浜0銆傛墍浠x瓒嬩簬1涓哄彲鍘婚棿鏂偣銆褰搙瓒嬩簬1+鏃舵瀬闄愯秼浜1锛屽綋x瓒嬩簬1-鏃舵瀬闄愯秼浜0銆傛墍浠瓒嬩簬1涓鸿烦璺冮棿鏂偣锛屽睘浜庣涓绫婚棿鏂偣銆傜瓟妗堝簲璇ラ塂
绛旓細璇佹槑锛氬洜涓x鈫1锛屾墍浠ュ彲浠ラ檺鍒秥x-1|<1/2锛屽垯1/2<x<3/2锛屽垯瀵逛簬浠绘剰灏忕殑姝ｆ暟e>0锛屾湁|(1/x)-1| =|x-1|/|x| <2|x-1|锛岃浣夸箣<e锛岄渶瑕亅x-1|<e/2锛屼簬鏄彧瑕佸彇d=min{1/2锛宔/2}>0锛屽氨鏈夊綋|x-1|<d鏃讹紝|(1/x)-1|<e鎴愮珛銆傝瘉姣曘
绛旓細濡傛灉鏋侀檺瀛樺湪 绛変环浜 宸﹀彸鏋侀檺鐩哥瓑 宸︽瀬闄 lim x->1- 1/(1-x) = 姝ｆ棤绌 鍥犱负x<1锛1-x>0 鑰屽彸鏋侀檺 lim x->1+ 1/(1-x)=璐熸棤绌 鍥犱负x>1锛1-x<0 鍥犱负宸﹀彸鏋侀檺涓嶇瓑锛屾墍浠ユ瀬闄恖imx->1 1/1-x涓嶅瓨鍦
绛旓細鎵浠ワ紝褰搙瓒嬭繎浜1鏃讹紝鏃犵┓灏1-x鍜1-x³鍚岄樁銆傚洜涓 lim锛坸鈫1锛1/2(1-x^2)/锛1-x锛=lim锛坸鈫1锛1/2(1-x)锛1锛媥锛/锛1-x锛=lim锛坸鈫1锛1/2(1锛媥锛=1/2脳2 =1 鎵浠ワ紝鏃犵┓灏1-x鍜岋紙1/2锛夛紙1-x²锛夊悓闃朵笖绛変环銆備袱涓棤绌峰皬鐩告瘮姹傛瀬闄愶紝鑻ユ瀬闄愮瓑浜1,鍒欏畠浠...
绛旓細x瓒嬩簬1鏃讹紝lnx鐨勭瓑浠锋棤绌峰皬鏄痻-1銆傚洜涓簂nx鐨勫鏁版槸1/x锛屽湪x=1鏃剁殑鍊兼槸1锛宭nx=1脳(x-1)+o(x)锛屼綘涔熷彲浠ョ洿鎺ユ眰lnx/(x-1)鍦▁瓒嬩簬1鏃跺鐨勬瀬闄愭槸1銆傛瀬闄愭濇兂鐨勬濈淮鍔熻兘 鏋侀檺鎬濇兂鍦ㄧ幇浠ｆ暟瀛︿箖鑷崇墿鐞嗗绛夊绉戜腑锛屾湁鐫骞挎硾鐨勫簲鐢紝杩欐槸鐢卞畠鏈韩鍥烘湁鐨勬濈淮鍔熻兘鎵鍐冲畾鐨勩傛瀬闄愭濇兂鎻ず浜嗗彉閲忎笌甯搁噺...
绛旓細瀵逛换鎰 蔚>0 ,瑕佷娇锛殀(x²-1)/(x-1)锛2| < 蔚 鎴愮珛,姝ゆ椂鍙锛殀(x²-1)/(x-1)锛2|锛潀x-1|
绛旓細1銆褰 x = 1锛浠ｅ叆鍚庯紝濡傛灉璁＄畻鍑烘潵鐨勬槸涓涓叿浣撶殑鏁板硷紝閭ｈ繖涓暟鍊煎氨鏄瓟妗堛.濡傛灉璁＄畻鍑烘潵鐨勭粨鏋滄槸鏃犵┓澶э紝浣嗏滄瀬闄愪笉瀛樺湪鈥濅簲涓瓧灏辨槸绛旀銆備絾鏄垜浠線寰鍙堣嚜鐩哥煕鐩惧湴鍐欎笂 lim銆併併= 鈭炪.2銆佸鏋滀唬鍏ュ悗锛屽彂鐜版槸涓嶅畾寮忥紝閭ｅ氨蹇呴』鐢ㄥ叾浠栨柟娉曡В绛斻.3銆佷笅闈㈢粰浣犻儴鍒嗘荤粨锛屽洜涓虹瘒骞呭法澶э紝鏃犳硶...
