absolutely integrability implies function approaches zero at positive infinity

Question

Is the following statement true?

$$\text{If function $f$ is absolutely integrable on $[0, \infty)$, this implies } \lim_{x \rightarrow \infty} f (x) = 0.$$

If yes then how would I prove it?

Note: I am considering the Riemann integrability and the function is $(f: \mathbb R \rightarrow \mathbb R)$.

Thanks and best regards

Answer 1

It is not true. Suppose $$ f(x) = \begin{cases} 1 & \text{if for some integer $n$, } n\le x \le n + \dfrac 1 {2^n}, \\ 0 & \text{otherwise}. \end{cases} $$ Then $$ \int_0^\infty f(x)\,dx = 2 < \infty, \text{ but }\limsup_{n\to\infty} f(x)=1. $$