L'Hospital's Rule may be applied more than once when evaluating

MATHEMATICS · Single correct · CBSE · Mathematics · Integrals · Integrals

Problem

\displaystyle\lim_{x\to a}\frac{f(x)}{g(x)}

. After differentiating once, which condition must still hold at

x=a

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}

(c)

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

and

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

(d)

g'(x)\neq0\,\text{ for all }x\text{ near }a

but

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