A function's absoulute integrability and its limit's relationship

Question

If a function $f(x)$ is defined on the real numbers and absolutely integrable, does it say anything about the $lim_{x\to\infty}xf(x)$ ?

Answer 1

This limit can exist, but if it does then it must be zero. Because if it weren't, then $|f(x)|\geq c/x$ for some $c>0$ and $$\int |f|\geq \int c/x=\infty.$$

It can also happen that this limit not exist, with limsup and/or liminf being infinite