Question

求极限！！！！

求极限！！！！如图，要有具体过程。

Answer 1

以上，请采纳。

追问

答案是e的-1/2次方

追答

不好意思，确实是e的-1/2次方，之前做的有点问题。

追问

怎么算

追答

额，我一起发了，可能系统有问题，你没看到，再给你发一次吧。

追问

3Q

Answer 2

let

y=1/x
consider
L = lim(x->∞) (1+1/x)^(x^2) / e^x
lnL
=lim(x->∞) [x^2.ln(1+1/x) -x]
=lim(y->0) [ln(1+y) -y] /y^2 (0/0)
=lim(y->0) [1/(1+y) -1] /(2y)
=lim(y->0) 1/[2(1+y)]
=1/2
=>
L = e^(1/2)
lim(n->∞) (1+1/n)^(n^2) / e^n = L = e^(1/2)

追问

Your answer is wrong.

The correct answer is e^(-1/2).

I tried my best, but finally I failed.

追答

中间出错， 不好意思！

let

y=1/x
consider
L = lim(x->∞) (1+1/x)^(x^2) / e^x
lnL
=lim(x->∞) [x^2.ln(1+1/x) -x]
=lim(y->0) [ln(1+y) -y] /y^2 (0/0)
=lim(y->0) [1/(1+y) -1] /(2y)
=lim(y->0) -1/[2(1+y)]
=-1/2
=>
L = e^(-1/2)
lim(n->∞) (1+1/n)^(n^2) / e^n = L = e^(-1/2)