What is the limit of (n/(n-1))^(n+2) as n approaches infinity?

[Telemachus]

Homework Statement

Hi there. I've found some difficulties on solving this limit:
$$\displaystyle\lim_{n \to{}\infty}{(\displaystyle\frac{n}{n-1})^{n+2}}$$

I thought of working with the function $$f(x)=(\displaystyle\frac{x}{x-1})^{x+2}$$
And then apply L'Hopital

This way:
$$\displaystyle\lim_{x \to{}\infty}{(\displaystyle\frac{x}{x-1})^{x+2}=e^{\displaystyle\lim_{x \to{}\infty}{(x+2) ln (\displaystyle\frac{x}{x-1})}}=e^{\displaystyle\lim_{x \to{}\infty}{\displaystyle\frac{(\displaystyle\frac{x}{x-1})}{\displaystyle\frac{1}{(x+2)}}}$$

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.

 

[jgens]

Note that we have ...

$$\frac{n}{n-1} = 1 + \frac{1}{n-1}$$

This gives us ...

$$\left(\frac{n}{n-1}\right)^{n+2} = \left(1+\frac{1}{n-1}\right)^{n-1}\left(1+\frac{1}{n-1}\right)^3$$

Evaluate the limit as $n \to \infty$ for each of the terms on right and then apply some "limit laws" to justify your result.

 

[Telemachus]

Thanks.