=e^{\lim_{n\to \infty } \frac{-2n^2}{n^2\left(1+\frac{2}{n}\sqrt{n^2+ 2n}+1+\frac{2}{n}\right)}}=e^{\lim_{n\to \infty } \frac{-2}{\left(1+2\sqrt{1+ \frac{2}{n}}+1+\frac{2}{n}\right)}}=e^{\frac{-2}{4}}=e^{-1/2}=\frac{1}{\sqrt{e}}