Question

求lim(x→∞)[x^(1+x)/(1+x)^x-x/e]极限

Answer 1

简单计算一下即可，答案如图所示

Answer 2

∵
lim
x^(1/x)
x→∞
=
lim
e^[lnx^(1/x)]
x→∞
=
lim
e^[(1/x)lnx]
x→∞
=
lim
e^[1/x]
=
1
x→∞
∴
lim
(1+x)^(1/x)
x→∞
∴
lim
(1+x)^{[1/(x+1)][(x+1)/x]}
x→∞
=
1^1
=
1
∴
lim
[(1+x)^(1/x)
-
e]/x
=
(1
-
e)/
∞
=
0
x→∞
楼主已经更正，按照新的极限要求，重新解答如下：
∵
lim
(1+x)^(1/x)
=
e
x→0
∴
lim
[(1+x)^(1/x)
-
e]/x
(0/0型)
x→0
=
lim
{[(1+x)^(1/x)][x/(1+x)
-
ln(1+x)]/x²
-
0}/1
x→0
=
lim
e[x/(1+x)
-
ln(1+x)]/x²
x→0
=
lim
e[x
-
(1+x)ln(1+x)]/(x²+
x³)
(0/0型)
x→0
=
lim
e[1
-
ln(1+x)
-
1]/(2x
+
3x²)
x→0
=
lim
-eln(1+x)/(2x
+
3x²)
(0/0型)
x→0
=
lim
-e/[(1+x)(2
+
6x)]
x→0
=
-e/2