Behavior of a squared-integrable function close to a specific value of $x$.

Question

Is there a way to prove that an $L^2(\mathbb{R})$ function $f$ close to $x=0$ is ${o}(x^{-1})$, in the sense that $$ \lim_{x\to 0} \left(|x{f(x)}|\right)=0? $$ I am working on a proof involving $L^2(\mathbb{R})$, and this fact would be really useful. It seems reasonable since the function needs to be integrable across $x=0$ but I am not exactly sure how to prove that. No need of the detailed proof, just an idea of what tool I should be using. This is an weaker condition then what I was asking in that other question. What I had as an answer was a clever example that disproved what I hoped was true. I hope the statement here is true. While I am still working on it, I have not found a way to prove it yet. I usually feel a little at a loss when wanting to prove things for $L^2$ functions, which can have strange behaviors. That's why I need help with that.