Chapter 2: Limits and Continuity - Section 2.4 - One-Sided Limits - Exercises 2.4 - Page 75: 33

$$\lim _{t \rightarrow 0} \frac{\sin (1-\cos t)}{1-\cos t} =1$$

Given $$\lim _{t \rightarrow 0} \frac{\sin (1-\cos t)}{1-\cos t} $$ Let $ x=1-\cos t \ \Rightarrow $ at $ t=0 $, we get $ x\rightarrow0$ So, we get \begin{aligned} L&=\lim _{t \rightarrow 0} \frac{\sin (1-\cos t)}{1-\cos t} \\ &= \lim _{x \rightarrow 0} \frac{\sin x}{x} \\ &=1 \end{aligned}