I want to find a function, such that the function shall be $p$-integrable in the sense of $\int_{\mathbb{R}}|f(x)|^pdx<\infty$ for any $p\in[1,\infty)$ and that this does NOT imply $\lim_{x\to\infty}f(x)=0$.
Now it's similar to the question here Continuous unbounded but integrable functions, but I'm not looking for a function which is unbounded in any point, just one which limit in infinity does not converge to $0$ or better say does not exist.
Does anyone know of any examples?