Question

求解极限。。。

哭了哭了
感觉好简单 就是不会做

Answer 1

用泰勒公式求解即可。过程如下

Answer 2

y->0+

(1+y)^(1/y)

=e^[ ln(1+y)/y]

泰勒公式 ln(1+y) = y -(1/2)y^2 +o(y^2)

=e^【[y-(1/2)y^2 +o(y^2)]/y】

=e^【[1-(1/2)y +o(y)]】

e/(1+y)^(1/y) = e^[(1/2)y+o(y)]

//

lim(x->+无穷) e^x/(1+1/x)^(x^2)

令y=1/x

=lim(y->0+) e^(1/y)/(1+y)^(1/y^2)

=lim(y->0+) [e/(1+y)^(1/y) ]^(1/y)

=lim(y->0+) { e^[(1/2)y] }^(1/y)

=e^(1/2)

得出结果

lim(x->+无穷) e^x/(1+1/x)^(x^2) =e^(1/2)