Question

求高等数学极限问题。谢谢大家了！！

Answer 1

罗比达法则：
先求y=(1+1/n)^2n的导数：
lny=2nln(1+1/n)
y'/y=2ln(1+1/n)+2n/(1+1/n)*(-1/n^2)
y'/y=2ln(1+1/n)-2/(n+1)
y'=[2ln(1+1/n)-2/(n+1)]*(1+1/n)^2n

lim n(e^2-(1+1/n)^2n)
=lim (e^2-(1+1/n)^2n)/(1/n)
=lim -{[2ln(1+1/n)-2/(n+1)]*(1+1/n)^2n}/(-1/n^2)

lim [ln(1+1/n)-1/(n+1)]/(1/n^2)
=lim [1/(1+1/n)*(-1/n^2)+1/(n+1)^2]/(-2/n^3)
=lim n^2/2(n+1)^2
=1/2.

所以：
lim n(e^2-(1+1/n)^2n)
=2*e^2*1/2

追问

你这是哪道题目的答案？不是我问的啊。