When evaluating the limit

MATHEMATICS · Single correct · AP · Calculus AB · Limits and Continuity · Limits and Continuity

Problem

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

, a student first rewrites

1-\cos^{2}x

\sin^{2}x

. Which of the following best explains how this step facilitates finding the limit? (a) It converts the numerator into a form involving

\dfrac{\cos x}{x}

, whose limit as

x\to 0

is known to be

1

. (b) It factors out a common

\sin x

term that cancels with the denominator, eliminating the

0/0

indeterminate form and leaving the familiar limit

\displaystyle \frac{\sin x}{x}

. (c) It shows that the fraction simplifies to

1

for every

x\neq 0

, so the limit must be

1

. (d) It allows the small-angle approximation

\sin x\approx x

to be applied immediately, giving

\dfrac{x}{x}=1

without further work.