Determine whether or not the following limit exists, and find its value if it exists: $$\lim \limits\_{n \to\infty}\ \left[n-\frac{n}{e}\left(1+\frac{1}{n}\right)^n\right] $$

I think the limit of $\left(1+\frac{1}{n}\right)^n$ is $e$, but I am not sure I can use this or not in the limit calculation. Could you please help me to solve this? Thank you!