Originally Posted by
Failure The standard trick here is to multiply with $\displaystyle \frac{\sqrt{n+1}+\sqrt{n}}{\sqrt{n+1}+\sqrt{n}}$:
$\displaystyle \lim_{n}(\sqrt{n+1}-\sqrt{n})=\lim_{n\to\infty}\frac{n+1-n}{\sqrt{n+1}+\sqrt{n}}=\lim_{n\to\infty}\frac{1}{ \sqrt{n+1}+\sqrt{n}}=0$
Well, let's try the same basic trick once more
$\displaystyle \lim_{n\to\infty}(\sqrt{n^2-n}-n)=\lim_{n\to\infty}\frac{n^2-n-n^2}{\sqrt{n^2-n}+n}=\lim_{n\to\infty}\frac{-n}{\sqrt{n^2-n}+n}$
$\displaystyle =\lim_{n\to\infty}\frac{-1}{\sqrt{1-\frac{1}{n}}+1}=-\frac{1}{2}$