Which of the following statements best explains why ?

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

Problem

Which of the following statements best explains why

\displaystyle \lim_{x\to\infty}\dfrac{\sin x}{x}=0

? (a) Because applying L’Hospital’s Rule to

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

yields

\displaystyle \lim_{x\to\infty}\frac{\cos x}{1}=0

, so the original limit is

0

. (b) Because

\sin x

is always between

-1

and

1

while

|x|

increases without bound, forcing

\dfrac{\sin x}{x}

to approach

0

. (c) Because

\dfrac{\sin x}{x}

oscillates only between

-1

and

1

for all

x

, so its limit must be

0

. (d) Because

\sin x

itself approaches

0

x\to\infty

, making the quotient

0