Chapter 3.6, Problem 70ES

Suppose that functions $f$ and $g$ are differentiable at $x=a$ and that $f\left(a\right)=g\left(a\right)=0$ . If $g^{\prime }\left(a\right)\ne 0$ , show that

$\underset{x\to a}{\lim }\frac{f\left(x\right)}{g\left(x\right)}=\frac{f^{\prime }\left(a\right)}{g^{\prime }\left(a\right)}$

without using L’Hopital’s rule.