Which of the following statements best justifies that ?

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

Problem

Which of the following statements best justifies that lim⁡x→0sin⁡xx=1\displaystyle \lim_{x\to 0}\frac{\sin x}{x}=1? (a) Since for very small angles (in radians) sin⁡x≈x\sin x\approx x, replacing sin⁡x\sin x by xx gives xx=1\dfrac{x}{x}=1, so the limit is 11. (b) Because sin⁡x\sin x is continuous at x=0x=0 and sin⁡0=0\sin 0=0, the ratio sin⁡xx\dfrac{\sin x}{x} is also continuous there, hence equals 00=1\dfrac{0}{0}=1. (c) Because sin⁡x\sin x is differentiable at x=0x=0 and ddx(sin⁡x)∣x=0=1\displaystyle \frac{d}{dx}(\sin x)\big|_{x=0}=1. (d) Applying the limit law lim⁡x→0sin⁡xlim⁡x→0x=00\displaystyle \frac{\lim_{x\to0}\sin x}{\lim_{x\to0}x}=\frac{0}{0} and canceling the zeros leaves 11, so the limit is 11.

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