$\lim\limits_{x\to+\infty}\frac{1}{\sqrt{n^{2}+2}-\sqrt{n^{2}+4}}$
$=\lim\limits_{x\to+\infty}\frac{1}{\frac{(\sqrt{n^{2}+2})^{2}-(\sqrt{n^{2}+4})^{2}}{\sqrt{n^{2}+2}+\sqrt{n^{2}+4}}}$
$=\lim\limits_{x\to+\infty}\frac{\sqrt{n^{2}+2}+\sqrt{n^{2}+4}}{2-4}$
$=\lim\limits_{x\to+\infty}\frac{n(\sqrt{1+\frac{2}{n^{2}}}+\sqrt{1+\frac{4}{n^{2}}})}{-2}$
$=\frac{n.(1+0+1+0)}{-2}$
$=-n$
$=-\infty$