The limit ( _ x 0 x x =1 )

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

Problem

The limit

\displaystyle \lim_{x\to 0}\frac{\tan x}{x}=1

. Which of the following statements best explains this result? (a) Because

\tan x

equals

\sin x

for every

x

, the ratio

\dfrac{\tan x}{x}

is the same as

\dfrac{\sin x}{x}

, whose limit is

1

. (b) Applying L’Hôpital’s Rule gives

\displaystyle \lim_{x\to 0}\dfrac{\tan x}{x}=\lim_{x\to 0}\dfrac{\sec^{2}x}{1}=0

, so the limit is

0

. (c) Writing

\tan x=\dfrac{\sin x}{\cos x}

and using

\sin x\approx x

and

\cos x\approx 1

for small

x

shows the ratio approaches

1

. (d) Because

\tan x

has a vertical asymptote at

\pi/2

, its ratio with

x

must stay finite and therefore equals

1

near

0