- #1
Telemachus
- 835
- 30
Homework Statement
Hi there. I've found some difficulties on solving this limit:
[tex]\displaystyle\lim_{n \to{}\infty}{(\displaystyle\frac{n}{n-1})^{n+2}}[/tex]
I thought of working with the function [tex]f(x)=(\displaystyle\frac{x}{x-1})^{x+2}[/tex]
And then apply L'Hopital
This way:
[tex]\displaystyle\lim_{x \to{}\infty}{(\displaystyle\frac{x}{x-1})^{x+2}=e^{\displaystyle\lim_{x \to{}\infty}{(x+2) ln (\displaystyle\frac{x}{x-1})}}=e^{\displaystyle\lim_{x \to{}\infty}{\displaystyle\frac{(\displaystyle\frac{x}{x-1})}{\displaystyle\frac{1}{(x+2)}}}[/tex]
And then I've applied L'hopital, but it didn't make the things easier. I've applied L'hopital unless two times. I don't know if what I did is right. And I think there must be an easier way of solving this.