# uniformly continuous and $\int_0^\infty f(t)\,\mathrm dt$ exists $\implies \lim_{x\to\infty}f(x) = 0$ [duplicate]

I appreciate your help with this one. Let $f \colon[0,\infty)\rightarrow \mathbb{R}$ be uniformly continuous and let the integral $\int_0^\infty f(t)\,\mathrm dt$ exist and be final.

I need to show that $$\lim_{x\to\infty}f(x) = 0.$$

Thank you very much.