$f:[a,\infty)\to\mathbb R$ be continuous such that $\int_a^\infty f(x)dx$ exists finitely , then $\lim_{x \to \infty}f(x)=0$ ?
Prove that $f$ continuous and $\int_a^\infty |f(x)|\;dx$ finite imply $\lim\limits_{ x \to \infty } f(x)=0$