I have seen counterexamples of continuous functions $f$ such that they are in $L^2(\mathbb{R})$ and $|f(x)|\rightarrow \neq 0$ as $|x|\rightarrow \infty$. Now I was wondering what would happen if we have a smooth function . Will we have in this case that $|f(x)|\rightarrow 0$ as $|x|\rightarrow \infty$?

Any help is appreciated, Thanks in advance.