Compute the limit $\lim_{n\to\infty}{(\sqrt[n]{e}-\frac{2}{n})^n}$ How can I compute this limit of the sequence?
 $$\lim_{n\to\infty}{(\sqrt[n]{e}-\frac{2}{n})^n}$$

This depends on the tools you master, but if you are aware that $$\sqrt[n]{\mathrm e}=\mathrm e^{1/n}=1+1/n+o(1/n),$$ then you can write that $$(\sqrt[n]{\mathrm e}-2/n)^n=(1-1/n+o(1/n))^n\to \mathrm e^{-1}.$$
