Chapter 2 - Section 2.4 - One-Sided Limits - Exercises - Page 85: 35

$$\lim_{\theta\to0}\frac{\sin\theta}{\sin2\theta}=\frac{1}{2}$$

$$\lim_{\theta\to0}\frac{\sin\theta}{\sin2\theta}=\lim_{\theta\to0}\frac{\sin\theta}{2\sin\theta\cos\theta}$$ $$\lim_{\theta\to0}\frac{\sin\theta}{\sin2\theta}=\lim_{\theta\to0}\frac{1}{2\cos\theta}$$ $$\lim_{\theta\to0}\frac{\sin\theta}{\sin2\theta}=\frac{1}{2\cos0}$$ $$\lim_{\theta\to0}\frac{\sin\theta}{\sin2\theta}=\frac{1}{2\times1}=\frac{1}{2}$$