How do I calculate this limit $\lim _{n\to \infty }\left(e^{\sqrt{n}}\left(1-\frac{1}{\sqrt{n}}\right)^n\right)$?

Question

How can I calculate this limit $$\lim _{n\to \infty }\left(e^{\sqrt{n}}\left(1-\frac{1}{\sqrt{n}}\right)^n\right)$$?

Answer 1

Hint:

Apply $\ln $ and then use Taylor's expansion of order $2$: $$\ln(1-u)=-u-\frac{u^2}2+o(u^2).$$

Answer 2

As Bernard commented $$A=e^{\sqrt{n}} \left(1-\frac{1}{\sqrt{n}}\right)^n$$ $$\log(A)={\sqrt{n}}+n\log\left(1-\frac{1}{\sqrt{n}}\right)$$ Now, using Taylor series $$\log\left(1-\frac{1}{\sqrt{n}}\right)=-\frac 1{\sqrt{n}}-\frac 1{2{n}}-\frac 1{3{n\sqrt{n}}}+O\left(\frac 1{n^2}\right)$$ $$\log(A)=-\frac 12-\frac 1{3{\sqrt{n}}}+O\left(\frac 1{n}\right)$$ Taylor again $$A=e^{\log(A)}=\frac{1}{\sqrt{e}}-\frac{1}{3 \sqrt{e}\sqrt{n}}+O\left(\frac{1}{n}\right)$$ which shows the limit and how it is approached.