Compute $\displaystyle\lim_{n \to \infty} \sqrt{n} \int_{-1}^{1} e^{\frac{-nx^{2}}{2}}f(x) \ \mathrm dx $ 
Compute $\displaystyle\lim_{n \to \infty} \sqrt{n} \int_{-1}^{1} e^{\frac{-nx^{2}}{2}}f(x) \ \mathrm dx $
where $f:[-1,1] \to \mathbb{R}$ is a continuous function.

I have no clue how to do it... Any hints?

Since $f$ is continuous on $[-1,1]$, the integrand is also continuous on $[-1,1]$. By mean-value theorem there is some $c \in [-1,1]$ such that the integral $= 2\exp (-nc^{2}/2)f(c)$, where $2$ is the length of the interval. Can you finish it from here?
