Suppose that $f$ is continuous on $[0,1]$. ($f'(x)$ may or may not exist).

How can I show that 

$$\lim\_{n\rightarrow\infty} \int\limits\_0^1 \frac{nf(x)}{1+n^2x^2} dx = \frac{\pi}{2}f(0)\;?$$

My attempt was to recognize that $\int\_0^1\frac{n}{1+n^2x^2}dx=\tan^{-1}(nx)$. But I can't simply use integration by parts on the original integral as $f'(x)$ might not exist. Any hints on how to solve this?

Help much appreciated...