Intuition for a function that belongs to a $L^p$ space

Question

Does a function $f(x):[0,\infty)\rightarrow R$ with $f\in L^p$ for $p<\infty$ have to die down to $0$ as $x\rightarrow \infty$? I some how feel that the $L^p$ norm exist only when the function dies down to $0$ and the rate at which it dies down to $0$ depends on $p$. Is this right?