<s>[INST] Determine whether or not the following sequence converges. If so, find the limit. Im not sure if I am evaluating the convergence of the following sequence correctly, and I am unsure of how to determine the limit. All help is greatly appreciated. 
$a_n = {1 \over (\sqrt{n^2-1}-\sqrt{n^2 +n})}$
My attempt at the convergence:
$ {1 \over (\sqrt{n^2-1}-\sqrt{n^2 +n})} < {1 \over n} $
$ \lim_{x\to \infty} {1 \over n} = 0$
Therefore, ${1 \over (\sqrt{n^2-1}-\sqrt{n^2 +n})}$ converges to as $ {n \to \infty} $
 [/INST] We have: $\sqrt{n^2-1} - \sqrt{n^2+n} = (\sqrt{n^2-1} - n) - (-n +\sqrt{n^2+n}) = -\dfrac{1}{\sqrt{n^2-1} + n} - \dfrac{1}{\sqrt{1+\dfrac{1}{n}} + 1} \to -\dfrac{1}{2}$. Thus $a_n \to -2$.
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