How to find this limit $\lim_{x\to+\infty}f(x)=0$

Question

let $f(x)\in C^{1}[0,+\infty)$ , and such improper integral $$\int_{0}^{+\infty}\left(|f(x)|+|f'(x)|\right)dx$$ is convergence

show that $$\lim_{x\to+\infty}f(x)=0$$

My try: since $$(e^xf(x))'=e^x[f(x)+f'(x)]$$ but this problem is $$|f(x)|+|f'(x)|$$ so I can't.Thank you

Answer 1

Since $\int |f'| < \infty$ and $f(t) = f(t_0) + \int_{t_0}^t f'(\tau) d \tau$ we have that $f_\infty = \lim_{t \to \infty} f(t) $ exists. Since $\int|f| < \infty$, we see that we must have $f_\infty =0$.