TITLE: Limit of $f(x)$ given that $ f(x)/x$ is known
QUESTION [6 upvotes]: 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.
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?

REPLY [2 votes]: Let $x_n \rightarrow 0$.
$y_n:= f(x_n)/x_n$, we have 
$y_n \rightarrow L.$
With
$f(x_n)=$
$(f(x_n)/x_n)(x_n)=(y_n)(x_n)$.
$\lim_{n \rightarrow \infty }f(x_n)=$
$\lim_{n \rightarrow \infty}((y_n)(x_n))=$
($\lim_{n \rightarrow \infty}(y_n))(\lim_{n \rightarrow \infty}(x_n))=$
$L \cdot 0=0.$