TITLE: Proving that a definition of e is unique
QUESTION [9 upvotes]: We can define $e$ as the number such that $\lim_{h \to 0} \frac{e^h-1}{h}=1$.  However, of course we can only define $e$ this way if it is unique, i.e., there is no other value $c$ for which that is a true statement. Could someone prove this uniqueness for me? I am tutoring a calculus student and he asked me something to this effect and I couldn't see why. Thanks!

REPLY [2 votes]: I have presented the theory of exponential and logarithmic functions in my blog post. For the current question we neeed to understand that for each $a>0$, the limit $$\lim_{h\to 0}\frac{a^{h}-1}{h}$$ exists and hence defines some function $f(a)$. Next we need to show that this function is strictly increasing and its range is whole of $\mathbb{R}$. This would imply that there is a unique number $e$ such that the above limit $f(e)=1$. The proof of these statements is bit long and it is well explained in the blog post linked above.