When evaluating

MATHEMATICS · Single correct · AP · Calculus AB · Contextual Applications of Differentiation · Contextual Applications of Differentiation

Problem

When evaluating

\displaystyle\lim_{x\to 0}\dfrac{\cos x-1}{x^{2}}

, a student differentiates numerator and denominator once to obtain

\displaystyle\dfrac{-\sin x}{2x}

and decides to differentiate again. Which of the following best justifies the need for the additional differentiation? (a)

-\sin x

tends to

-1

while

2x

tends to

0

, so the quotient becomes unbounded; differentiating again may yield a finite limit. (b) Because the denominator contains

x^{2}

, L’Hospital’s Rule must be applied twice—once for each power of

x

. (c) The new quotient is still indeterminate because both

-\sin x

and

2x

approach

0

x\to 0

. (d) A second differentiation guarantees the denominator’s order decreases faster than that of the numerator, ensuring convergence.