Given that $$ \lim\_{x \to 0} \dfrac{f(x)}{x} $$ exists as a real number, I am trying to show that $\lim\_{x\to0}f(x) = 0$. There is a similar question here:
[Limit of f(x) knowing limit of f(x)/x](https://math.stackexchange.com/questions/1232577/limit-of-fx-knowing-limit-of-fx-x).

But this question starts with the assumption of $$ \lim\_{x \to 0} \dfrac{f(x)}{x} = 0, $$ and all I am assuming is that the limit is some real number. So the product rule for limits doesn't really work here.

Or do I need to show that $$ \lim\_{x \to 0} \frac{f(x)}{x} = 0 $$ and then apply the product rule?