Can an improper integral with infinity singularity converge for any integrand?

Question

I have an integral $\int_0^\infty f(x) dx$, where $\lim_{x \rightarrow 0} f(x) = \infty$ and $\forall x \geq 0: f(x)>0$. Is there any $f(x)$ that fulfills these requirements such that the integral converges? If not, how can you prove it does not exist?