Mathematics

Problem

L’Hospital’s Rule may be applied more than once when evaluating $ \displaystyle\lim_{x\to a}\dfrac{f(x)}{g(x)}

. After differentiating once, which condition must still hold at

x=a$ in order to justify taking derivatives a second time? a) Both

f''

and

g''

exist and are continuous at

x=a

b) The quotient

\dfrac{f'(x)}{g'(x)}

is again an indeterminate form

\tfrac{0}{0}

\tfrac{\infty}{\infty}

\displaystyle\lim_{x\to a}f'(x)=0

and

\displaystyle\lim_{x\to a}g'(x)=0

g'(x)\neq 0\,\text{ for all } x \text{ near } a

but

\displaystyle\lim_{x\to a}f'(x)=0