I was asked in today's exam to prove/disprove the following: ($f$ is a continuous function in $[0,1]$)
$$\lim_{n \rightarrow \infty} \int_{0}^{1} \left(\frac{n f(x)}{1 + n^2x^2} \right) dx = \frac \pi 2 f(0)$$
I tried out a few functions $f(x)$, and decided that it was probably true. But all I could do was to reduce the integral to
$$\lim_{n \rightarrow \infty} \int_{0}^{1} \left(\frac{n f(x)}{1 + n^2x^2} \right) dx = \lim_{n \rightarrow \infty} \int_{0}^{n} \frac{f(x/n)}{1 + x^2} dx$$
which didn't lead me anywhere.