Evaluate the limit $\lim_{n\to\infty}\left[n+{n^2}\log{\frac{n}{n+1}}\right]$ Evaluate
$$\lim_{n\to\infty}\left[n+{n^2}\log{\frac{n}{n+1}}\right]$$
I know that this limit is equal to $\frac1{2}$ but I don't know how to do it.
Thanks!

Let $n=1/x
$$$\lim_{n\to\infty}\left[n-{n^2}\log{\frac{n+1}{n}}\right]=\lim_{x\to0}\left[\frac{x-\log(1+x)}{x^2}\right]=\lim_{x\to0} \frac{1}{2(x+1)}=\frac{1}{2}$$
where the penultimate equality is obtained by  l'Hôpital's rule.
Note: as Marvis well noticed, the first equality is true because $\lim_{x\to0}\left[\frac{x-\log(1+x)}{x^2}\right]$ exists. 
Chris.
