Let $~f~$ be a non-negative continuous function on $~\mathbb{R}~$ such that $~\int_{0}^{∞} f(x)~dx~$ is finite,then $~\lim_{x\to ∞} f(x) = 0~$.
Prove that $f$ continuous and $\int_a^\infty |f(x)|\;dx$ finite imply $\lim\limits_{ x \to \infty } f(x)=0$