Evalute $\lim\limits_{n\to\infty}\int _{0}^{1}\frac{nf(x)}{1+n^2x^2}\,dx$
With the change $y=nx$ you get $$\int _{0}^{1}\frac{nf(x)}{1+n^2x^2}\,dx = \int _{0}^{n}\frac{f(\frac{y}{n})}{1+y^2}\,dx.$$ Now Lebesgue theorem should allow you to conclude.
With the change $y=nx$ you get $$\int _{0}^{1}\frac{nf(x)}{1+n^2x^2}\,dx = \int _{0}^{n}\frac{f(\frac{y}{n})}{1+y^2}\,dx.$$ Now Lebesgue theorem should allow you to conclude.