a function that is in $L^2$

Question

Can anyone give me an example of $f(x) $ such that $ f \in L^2 ( \mathbb R)$ but $ x^{\frac{1}{2}} f \notin L^1 ( \mathbb R ) $. Thanks!

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It seems that $f(x) = x^\alpha$ doesn't work...

Answer 1

Take $f(x)=x^{-1}\chi_{[1,\infty]}$