Suppose that $a>0 .$ Tangent lines to the exponential functions $f(x)=a^{x}$ are investigated in this exercise.
a. Use the formula for the slope of a tangent line to show that the slope of the tangent line to $f$ at $(c, f(c))$ is $$f^{\prime}(c)=a^{c} \lim _{h \rightarrow 0} \frac{a^{h}-1}{h}.$$
b. Show that the slope of the tangent line to $f$ at (0,1) is $$\lim _{h \rightarrow 0} \frac{a^{h}-1}{h}.$$
c. This limit cannot be computed by substituting $h=0$ in the expression because that results in the meaningless expression $0 / 0 .$ We will learn how to compute this limit in Section 3.6. For now, we will investigate it numerically. To be specific, let $a=2$. To identify the limit, graph the function $h \mapsto\left(2^{h}-1\right) / h$ in a window whose horizontal range is a small interval centered at 0. From the graph (zooming in if necessary), identify $$\lim _{h \rightarrow 0} \frac{2^{h}-1}{h}$$ to four decimal places.
d. Graph $f(x)=2^{x}$ tangent lines to the graph of $f$ at (0,1) and (2,4) in the same viewing window.
e. Repeat parts (c) and (d) with $a=3$. Use $(-1,1 / 3)$ and (1,3) as the points of tangency.