Let $f:[0,\infty)\to\mathbb{R} \ $ be a continuous function.
Suppose $\int_{0}^{\infty}f(x)dx\ $ converges,
Compute: $$\lim\limits_{n\to\infty}\int_{0}^{1}f(nx)dx.$$
My attempt:
I took $t=nx$, then $dt=n\ dx$, so we get:
$$\lim\limits_{n\to\infty}\int_{0}^{1}f(nx)dx=\lim\limits_{n\to\infty}\frac{1}{n}\int_{0}^{n}f(t)dt=0\cdot\int_{0}^{\infty}f(x)dx=0.$$
Is it true? Am I missing something?