lim2x为什么可变成2limx lim(x→∞)√x∧2+2x+2/x为啥是-1，能写下具体...

\u4e3a\u4ec0\u4e485\u9898\u7b54\u6848\u662f\u8fd9\u4e2a\uff1f\u53e6\u5916\u8bf7\u95eelim\u4e00\u4e2a\u5e38\u6570=\uff1f\uff1f limx=? lim2x=\uff1f\uff1f

\u8fd9\u4e2a\u6ca1\u6709\u5177\u4f53\u9898\u76ee\u662f\u65e0\u6cd5\u89e3\u7b54\u5f97\uff0c\u53ef\u4ee5\u624b\u673a\u62cd\u7167
\u770b\u5230\u9898\u76ee\u5c31\u53ef\u4ee5\u4e86\uff0c\u8fd8\u662f\u5efa\u8bae\u4f5c\u4e1a\u8fd8\u662f\u81ea\u5df1\u505a\uff0c
\u4e0d\u4f1a\u5f97\u53ef\u4ee5\u95ee\u95ee\u540c\u5b66\uff0c\u76f8\u4e92\u8fdb\u6b65
\u5e0c\u671b\u80fd\u5e2e\u5230\u4f60\uff0c\u8bf7\u91c7\u7eb3\u6b63\u786e\u7b54\u6848.
\u4f60\u7684\u70b9\u8d5e\u6216\u91c7\u7eb3\u662f\u6211\u7ee7\u7eed\u5e2e\u52a9\u5176\u4ed6\u4eba\u7684\u52a8\u529b

\u4e0a\u4e0b\u540c\u9664x
\u56e0\u4e3ax\u2192-\u65e0\u7a77
\u6240\u4ee5\u4e0a\u9762\u90a3\u4e2a\u6839\u53f7\u5728\u5316\u7b80\u65f6\uff0c\u9700\u8981\u5bf9\u7ed3\u679c\u53d6\u8d1f
\u5373
limx\u2192\u8d1f\u65e0\u7a77 \u221a\uff08x^2+2x+2\uff09/x
=limx\u2192\u8d1f\u65e0\u7a77 -\u221a\uff081+2/x+2/x^2\uff09/1
=-1

见下图（极限运算法则）：

lim2x涓轰粈涔堝彲鍙樻垚2limx
绛旓細瑙佷笅鍥撅紙鏋侀檺杩愮畻娉曞垯锛夛細

limx鈫 鏃犵┓(2x涓夋鍏➖x➕1)涓轰粈涔鏋侀檺涓2
绛旓細绛旀濡傚浘鎵绀

limx瓒嬪悜浜庢鏃犵┓2x/(x+2)涓轰粈涔绛変簬2
绛旓細lim 2x/(x+2)x鈫+鈭 =lim 2/(1+ 2/x)x鈫+鈭 =2/(1+0)=2/1 =2

limx鈫 鏃犵┓(2x涓夋鍏➖x➕1)涓轰粈涔鏋侀檺涓嶄负2
绛旓細y=锛2x³-x鍗1锛/锛坸³+ax²+bx+c锛夛紝鎵嶆槸鏋侀檺涓2.

涓轰粈涔坙im(2x²/x²+x)=2(x瓒嬭繎浜0),鑳借В閲婁竴涓嬩箞
绛旓細杩欎釜寮忓瓙鏄笉瀵圭殑 x->0鏃 lim[2x²/(x²+x)]=lim[2x/(x+1)]=0 x->鏃犵┓鏃 lim[2x²/(x²+x)]=lim[2/(1+1/x)]=2

涓轰粈涔坙im_{x鈫0}(e^{2x}-1)/x=lim[(2x)/x]
绛旓細瑙ｆ瀽锛歺鈫0鏃讹紝e^(2x)-1鍜2x鏄瓑浠锋棤绌峰皬

lim(2x+鈻硏)=2?(褰搙瓒嬪悜x0)
绛旓細lim(2x+鈻硏)=lim2x+lim鈻硏=2limx+lim鈻硏==0+2 x->x0 x0=0 鈻硏=2 瀵逛粯妤间富杩欐牱鐗涙瘮鐨勪笉鍙兂璞＄殑棰樺彧鑳借繖鏍峰洖绛斾簡锝烇綖

涓轰粈涔杩欓亾楂樻暟棰樻垜閿欎簡?
绛旓細濡傛灉涓嬮潰寮忓瓙鎴愮珛锛岃繕鏈夊繀瑕佺爺绌舵棤绌峰ぇ/鏃犵┓澶т笉瀹氬瀷涔堬紵杩欒偗瀹氫笉鎴愮珛鍟

lim(鈭垰tant)
绛旓細瀵瑰垎瀛愬垎姣嶅垎鍒眰瀵兼湁锛lim2x(tanx-x)/(tanx-sinx)=lim2x(tanx-x)/(1/2x^3)=lim4(tanx-x)/x^3 =lim4(tanx-x)/x^2=lim2(sec^2x-1)/x=2lim(tanx)^2/x=4limtanx*(secx)^2=0

楂樼瓑鏁板姹傛瀬闄愨︹
绛旓細涓嶆噦鍐嶉棶锛屾槑鐧借閲囩撼锛

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