Mathematics

Problem

A student wishes to compute

\displaystyle\lim_{x\to 0}\frac{\tan x}{x}

and first rewrites

\tan x

\dfrac{\sin x}{\cos x}

. Which of the following best explains why this rewrite can simplify the evaluation of the limit? a) Because after rewriting, the

x

in the denominator cancels with the

x

inside

\sin x

, leaving

\dfrac{1}{\cos x}

, whose limit as

x\to 0

0

. b) Because expressing

\tan x

\dfrac{\sin x}{\cos x}

turns the limit into a

0/0

indeterminate form to which L’Hospital’s Rule can be applied, giving

\sec^{2}0=1

. c) Because

\dfrac{\sin x}{x}

and

\cos x

each have well-known limits as

x\to 0

, enabling the expression to be factored into familiar pieces. d) Because

\dfrac{\sin x}{\cos x}

is continuous at

x=0

, so the entire expression is continuous there and the limit equals its value at

x=0