The limit . Which of the following statements best JUSTIFIES this…

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

Problem

The limit

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

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

\sin x\le x

for all small

x

, we have

\dfrac{\sin x}{x}\le 1

. Since the ratio cannot exceed

1

and gets arbitrarily close to it, the limit must be

1

. (b) Near

x=0

\cos x\le\dfrac{\sin x}{x}\le 1

, and both bounding functions approach

1

x\to 0

. (c) Using the linear approximation

\sin x\approx x

for very small

x

, we obtain

\dfrac{\sin x}{x}\approx\dfrac{x}{x}=1

, so the limit is

1

. (d) Because

\dfrac{\sin x}{x}

equals the derivative of

\sin x

x=0

(which is

\cos 0=1

), the limit must be

1