Evaluating a limit given the limit of the derivative as $x$ approaches infinity without using L’Hopital’s

I have been asked to evaluate the following limit as a challenge problem in my real analysis class. This was part of a problem set which mainly involved L’Hopital’s rule, but since you can’t use it here, I’m totally stuck. What theorems would I even use?

PROBLEM

Suppose $f: (1,\infty) \mapsto R \ $is differentiable on its domain. Also suppose that the limit of $f’(x)$ as $x $approaches infinity is zero.

Explain how one would find the value of $$\lim_{x\to\infty} \frac{f(x)}{x}$$