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.
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