Is the integral finite if the integrand is $o(x^{-1})$?

Question

According to theorem 2.2 in this file http://www.stat.umn.edu/geyer/old06/5101/notes/n1.pdf

If $\lim_{x\to\infty} \frac{g(x)}{x^{-1}} =0$, nothing can be said about the existence of $\int_a^\infty g(x)dx$.

However I think this integral should also be finite. Since I guess under this condition we can find an $\alpha < -1$ such that $g(x)$ has the same rate as $x^\alpha$. Then use the first case of this theorem, we can conclude the integral is finite.

If I'm wrong, can someone give a counterexample?

Answer 1

$$\int_e^\infty\frac{dx}{x\log x}=\log\log x\bigg|_e^\infty=\infty$$