I got stuck at limit problem 
$$\lim_{n\to\infty} n\bigg[e^{\frac x{\sqrt n}}-\frac x{\sqrt n}-1\bigg] = \frac{x^2}{2}$$

I'm not sure how to solve it.
hope somebody could help me!
_ Is there any way to see solutions for limit problem?

Sice $e^{z/\sqrt{n}}$ is analytic we can consider
$$
e^{z/\sqrt{n}} = 1 + \frac{z}{\sqrt{n}} + \sum_{m=2}^\infty \frac{z^m}{\sqrt{n^m}m!}
$$
So
$$
n\left(e^{z/\sqrt{n}} - 1 - \frac{z}{\sqrt{n}}\right) = \sum_{m=2}^\infty \frac{z^n}{n^{m/2-1}m!} = \frac{z^2}{2} + \sum_{m=3}^\infty \frac{z^m}{n^{m/2-1}m!}
$$
Taking limit and changing limit and sum (you can check easily dominate convergence) have the result.
