$f$ real-valued function that dies of in infinity but $f^p$ not integrable for any $p$.

Question

Is there a positive continuous function on $\mathbb R$ such that $f(x) \to 0$ as $x \to \pm \infty$ but $f^p$ not integrable for any $p>0$?

Answer 1

How about $$ f(x)=\frac{1}{1+\log(1+x^2)}? $$