Recently I started considering derivative (calculating it by definition) of an exponential function $a^x$. It requires couple of additional steps, one of them is to define constant $e$ as the only real number that satisfies:
$$ 1 = \lim_{h\to 0} \frac{a^h-1}{h} $$
And here I am stuck. Why such definition is correct? First we must prove that fot every real number $r$ exists such $a$ that
$$ r = \lim_{h\to 0} \frac{a^h-1}{h} $$
And I do not know how to proceed with it. Any suggestions? Concept of proof?
The online textbook CLP-1 Differential Calculus (by Feldman/Rechnitzer/Yeager) develops the derivative of exponential functions in exactly this way; they include a proof that there is a unique such number in Section 2.7 (especially Section 2.7.2).