real analysis - Limit $limlimits_{nto +infty} e^{sqrt n }cdotleft(1 - frac1{sqrt n}right)^n$

$$\lim_{n\to +\infty}\ e^{\sqrt n }\cdot\Big(1 - \frac{1}{\sqrt n}\Big)^n$$

Answers are:

A)0

B)1

C)e

D)${\sqrt e}$

E)$\frac{1}{\sqrt e}$

I tried working on the second part to get it in a better form. In the end I got $e^{- \sqrt n}$. Returning at the beginning with this new form it would be:

$$\lim_{n\to +\infty}\ e^{\sqrt n }\cdot e^{- \sqrt n}$$ which would eventually turn into $$\lim_{n\to +\infty}\ e^{0}$$ so the answer would be B) 1 but it's not. The answer is E)$\frac{1}{\sqrt e}$ but I can't figure it out why.

Answer

Consider $$A= e^{\sqrt n } * \left(1 - \frac{1}{\sqrt n}\right)^n$$ Take logarithms of both sides $$\log(A)=\sqrt n+n\log\left(1 - \frac{1}{\sqrt n}\right)$$ Now, use the fact that for small values of $x$, $\log(1-x)=-x-\frac{x^2}{2}+O\left(x^3\right)$. Replace $x$ by $\frac{1}{\sqrt n}$ which makes $$\log(A)=\sqrt n+n(-\frac{1}{\sqrt n}-\frac{1}{2 n}+\cdots)$$ Expand and simplify.

I am sure that you can take from here.