real analysis - Defining the number $e$

In this YouTube video it is said that $e$ naturally arises as a number that allows us to take the derivatives of functions like $a^x$. So $e$ is defined as a number for which:
$$(e^x)'=e^x\lim_{h\rightarrow 0}\frac {e^h-1}{h}=e^x$$

So essentially the limit should be $1$ for this to happen. How would you prove that there is indeed such a real number, and find its value?

P.S.Please, don't say that there are other ways to define $e$ - I know that. It's just this particular definition that seems natural and interesting to me.