Counterexample on the limit of $\frac{f(x)}{x}$

Question

Is the following statement true or false?

$f$ is defined on the set of all real numbers, such that $\lim \limits_{x\to 0} \dfrac{f(x)}{x}$ is a real number $L$ and $f(0)=0$. Then $L=0$?

I can't draw up any counterexample.

Would be grateful for hint.

Answer 1

Simply set $f(x)=x$. Then $L=1$.

Answer 2

Hint: Simply let $f(x)=x^2+x$

Answer 3

Take $f=\sinh(x)$ while $L=1$ and $f(0)=0$

Answer 4

Infact $L$ can be any real number because $$\lim\limits_{x\to 0}\frac{\sin ax}x=a$$