Chapter 2: Limits and Continuity - Additional and Advanced Exercises - Page 104: 25

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\begin{align*} \lim _{x \rightarrow 0}\frac{\sin (1-\cos x)}{x}&=\lim _{x \rightarrow 0} \frac{\sin (1-\cos x)}{1-\cos x} \cdot \frac{1-\cos x}{x} \cdot \frac{1+\cos x}{1+\cos x}\\ &=\lim _{x \rightarrow 0} \frac{\sin (1-\cos x)}{1-\cos x} \cdot \lim _{x \rightarrow 0} \frac{1-\cos ^{2} x}{x(1+\cos x)}\\ &=1 \cdot \lim _{x \rightarrow 0} \frac{\sin ^{2} x}{x(1+\cos x)}\\ &=\lim _{x \rightarrow 0} \frac{\sin x}{x} \cdot \frac{\sin x}{1+\cos x}\\ &=1 \cdot\left(\frac{0}{2}\right)\\ &=0 \end{align*}