There exists a sequence of real numbers $(x_n)$ such that $\lim_{n \rightarrow \infty}x_n = \infty , \lim_{n \rightarrow \infty}x_nf(x_n)=\infty.$

Question

Suppose $f:\mathbb{R} \rightarrow \mathbb{R}$ is a continuous function such that $$\int_{-\infty}^{\infty}|f(t)|dt<\infty.$$ Prove that there exists a sequence of real numbers $(x_n)$ such that $$\lim_{n \rightarrow \infty}x_n = \infty , \lim_{n \rightarrow \infty}x_nf(x_n)=\infty.$$

This is one of my past exam questions. I have no idea how to start at all. Any hint would be appreciated.