Suppose that functions f and g are differentiable at x = aand that f(a) = g(a) = 0. If

Chapter 3, Problem 70

(choose chapter or problem)

Suppose that functions f and g are differentiable at \(x=a\) and that \(f(a)=g(a)=0\). If \(g^{\prime}(a) \neq 0\)

, show that \(\lim _{x \rightarrow a} \frac{f(x)}{g(x)}=\frac{f^{\prime}(a)}{g^{\prime}(a)}\) without using L'Hopital's rule. [Hint: Divide the numerator and denominator of \(f(x) / g(x)\) by \(x-a\) and use the definitions for \(f^{\prime}(a) \text { and } g^{\prime}(a)\)

Equation Transcription:

Text Transcription:

x=a

f(a)=g(a)=0

g prime (a) neq 0

lim _x right arrow a f(x)/g(x) = f prime (a)/g prime (a)

f(x)/g(x)

X-a

f prime (a) and g prime (a)