$\liminf_{|x|\to \infty} |x f(x)| =0$ for a Lebesgue integrable function on $\mathbb{R}$

Question

Having trouble proving something I saw on an old qualifying exam. It goes as follows, let $f: \mathbb{R} \to \mathbb{R}$ be a Lebesgue integrable function such that $$ \int f(x) \: \textrm{d}x < \infty , $$ then $$ \liminf_{|x| \to \infty } |x f(x) | = 0$$
I am not sure how to proceed as I have not seen anything of this form before, well there is a nice analogue in the Riemann case dealing with limits, I am sure what properties of Lebesgue integrable function to use in this case. Thanks.

Answer 1

If $\liminf_x |xf(x)| >0$ then there is some $L$, $\delta>0$ such that for $|x| \ge L$ we have $|xf(x)| \ge \delta$.

Then $|f(x)| \ge {\delta \over |x|}$ for $|x| \ge L$ and hence $f$ is not integrable.