After summarizing a few examples, I induce a guess:

$f(x)>0$ is a continuous function in $(0, +infty)$, $int_0^{+infty}frac{mathrm{d}x}{f(x)}$ converges, then $limlimits_{lambdarightarrow +infty}frac{1}{lambda}int_0^{lambda}f(x)mathrm{d}x=+infty$.

I believe it is correct, but I have no idea how to prove it. Thanks a lot if you can provide a valid proof or some hints.