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

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