How to determine the convergence behavior of the sequence $x_{n}=\sqrt{n^{2}+11n+21}-\sqrt{n^{2}+6}$? How to determine the convergence behavior of the sequence $x_{n}=\sqrt{n^{2}+11n+21}-\sqrt{n^{2}+6}$ and its limit?
My effort: I don't know where to begin. 

Hint: multiply and divide $x_n$ by $\sqrt{n^2+11n+21}+\sqrt{n^2+6}$
$\sqrt{n^2+11n+21}-\sqrt{n^2+6}=
(\sqrt{n^2+11n+21}-\sqrt{n^2+6})({\sqrt{n^2+11n+21}+\sqrt{n^2+6}\over {\sqrt{n^2+11n+21}+\sqrt{n^2+6}}}$
$={{11n+15}\over{\sqrt{n^2+11n+21}+\sqrt{n^2+6}}}$. So the limit is $11/2$.
