Limit exists or not? $lim limits_{n toinfty} left[n-frac{n}{e}left(1+frac{1}{n}right)^nright] $

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!

Answer

Note that we can write

$$\begin{align}
n-\frac ne\left(1+\frac1n\right)^n&=n-\frac ne e^{n\log\left(1+\frac1n\right)}\\\\
&=n-\frac ne e^{n\left(\frac1n -\frac{1}{2n^2}+O\left(\frac{1}{n^3}\right)\right)}\\\\
&=n-n\left(1-\frac{1}{2n}+O\left(\frac{1}{n^2}\right)\right) \\\\
&=\frac12+O\left(\frac1n\right)
\end{align}$$