Evaluating $\lim_{n \to \infty} n\left(1-\frac{1}{e}\left(1+\frac{1}{n}\right)^{n} \right)$

Question

$$\lim_{n \to \infty} n\bigg(1-\dfrac{1}{e}\bigg(1+\dfrac{1}{n}\bigg)^{n} \bigg)$$

If I write expansion of $\bigg(1+\dfrac{1}{n}\bigg)^{n}$ it was equal to expansion of $e$ so $n-n=0$. Is limit is zero ?

Edit: Uploading screen shot of my response. They marked it correct and I don't know if answer key is wrong or not ? Please can anyone say for sure like with $100$ percent surety if answer given is wrong.