I want to prove that if $$\lim_{h \to 0} \frac{a^h-1}{h}=1$$ then $a$ must equal Euler's constant, denoted as "$e$." However, I have some specific constraints for this proof:
1.$e$ is defined only with the limit definition ($\left(1+\frac{1}{x}\right)^x$ as x tends to infinity).
- I only want to start the proof by manipulating $\lim_{h \to 0} \frac{a^h-1}{h}=1$ and I don't immediately want to start with the limit definition of e. I want to solve it as if I just discovered the limit and did not know that $a=e$