Which of the following statements most directly supports the conclusion that ?

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

Problem

Which of the following statements most directly supports the conclusion that lim⁡x→0sin⁡xx=1\displaystyle\lim_{x\to 0}\frac{\sin x}{x}=1? (a) For xx near 00, 1≤sin⁡xx≤1cos⁡x1\le \dfrac{\sin x}{x} \le \dfrac{1}{\cos x}, and both endpoint expressions approach 11 as x→0x\to 0. (b) For xx near 00, cos⁡x≤sin⁡xx≤1\cos x \le \dfrac{\sin x}{x} \le 1, and lim⁡x→0cos⁡x=1\displaystyle\lim_{x\to 0}\cos x =1. (c) For xx near 00, 0<sin⁡xx<10< \dfrac{\sin x}{x} < 1, and lim⁡x→01=1\displaystyle\lim_{x\to 0}1 = 1. (d) For xx near 00, sin⁡x<x<tan⁡x\sin x < x < \tan x, and lim⁡x→0sin⁡x=lim⁡x→0tan⁡x=0\displaystyle\lim_{x\to 0}\sin x = \lim_{x\to 0}\tan x =0.

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