Q:

Problem with evaluating $\lim_{n \rightarrow \infty} e^{-n-t\sqrt{n}}\cdot \left(e^{e^{\frac{t}{\sqrt{n}}}\cdot n}-1\right)$

As above, I have a problem with evaluating
$$\lim_{n \rightarrow \infty}  e^{-n-t\sqrt{n}}\cdot \left(e^{e^{\frac{t}{\sqrt{n}}}\cdot n}-1\right).$$
I checked the result in the limit calculator and it should be equal to $e^{\frac{t^2}{2}}$. I have no idea how to obtain it.
My attempt:
$ \lim_{n \rightarrow \infty}  e^{-n-t\sqrt{n}}\cdot(e^{e^{\frac{t}{\sqrt{n}}}\cdot n}-1)=\lim_{n \rightarrow \infty} e^{-n-t\sqrt{n}+e^{\frac{t}{\sqrt{n}}}\cdot n}-\lim_{n \rightarrow \infty}e^{-n-t\sqrt{n}}=\lim_{n \rightarrow \infty} e^{-n-t\sqrt{n}+e^{\frac{t}{\sqrt{n}}}\cdot n}-0=?.$ 
Unfortunately I don't know how to proceed later, because I'm obtaining the limit of the type $0\cdot \infty$ which is of course problematic. 
Thanks for help in advance.

A:

Hint use Taylor expansion of $\displaystyle e^{\frac{t}{\sqrt{n}}}$.
$$\exp\left(\frac{t}{\sqrt{n}}\right) \approx 1 + \frac{t}{\sqrt{n}} + \frac{t^2}{2n} + O\left(\frac{t^3}{n^{3/2}}\right)$$

