Let $f(x)$ be a continuous function defined for $x \geq 0$ such that $f(x) \geq 0$ for all $x \geq 0$. Additionally, suppose that
$$ \lim_{x\to\infty} f(x) =a$$
where $a$ is a finite constant.
Is this information sufficient to prove that the improper integral
$$ \int_{0}^{\infty} f(x) \, dx \ $$
diverges, i.e.,
$$ \int_{0}^{\infty} f(x) \, dx = +\infty ? $$