Prove or disprove: if $\int_1^\infty f(x)\,dx$ converges, then $\lim\limits_{x\to\infty}f(x) =0$.
Is there a continuous function such that $\int_0^{\infty} f(x)dx$ converges, yet $\lim_{x\rightarrow \infty}f(x) \ne 0$?