If $\forall \epsilon >0, f(x) = \mathscr O(x^{1+\epsilon})$ is $\int^\infty \frac {1}{f(x)} = \infty$?

Question

If $f(x)$ is positive and $\forall \epsilon >0, f(x) = \mathscr O(x^{1+\epsilon})$ as $x \rightarrow \infty$, is $\int^\infty \frac {1}{f(x)} = \infty$ ?

For example, this holds for $f(x)= x \text ln(x)$ and $f(x)= x \sqrt{\text ln(x)}$.

Answer 1

Take $f(x) = x\ln^2 x$, as a counter example. (Or even $f(x) = x\ln x (\ln\ln x)^2$ if you really want to live on the edge.)