Is it necessarily true that $\lim\limits_{x \rightarrow \infty} f(x) = 0$?

Question

Suppose $f : \Bbb R \longrightarrow \Bbb R$ be a continuous function such that $\int\limits_{0}^{\infty} f(x)\ dx$ exists finitely. If $\lim\limits_{x \rightarrow \infty} f(x)$ also exists finitely then is it necessarily true that $\lim\limits_{x \rightarrow \infty} f(x) = 0$?

Any suggestion regarding this will be highly appreciated. Thank you very much for your valuable time.

Answer 1

If $L=\lim_{x \to \infty} f(x) >0$ then there exists $M$ such that $f(x) >\frac L 2$ for $x \geq M$ and $\int_M^{\infty} f(x)dx =\infty$. Similarly $L <0$ leads to a contradiction, so $L=0$.