Understanding the proof of derivative of exponential function.

Question

The following excerpt is from Stewart Calculus: Early Transcendentals, 7th Edition.

My question is how does the author arrive at:

$$\lim_{h \to 0}\frac{a^h-1}{h}=f'(0)$$

Answer 1

As per the author's work, we have:

$$f'(x) = a^{x} \lim_{h \to 0} \frac{a^h - 1}{a}$$

Replacing $x = 0$ in the above expression yields:

$$f'(0) = a^0 \lim_{h \to 0} \frac{a^h - 1}{a} = \lim_{h \to 0} \frac{a^h - 1}{a}$$

as desired.