Chapter 1 - Limits and Continuity - 1.6 Continuity of Trigonometric Functions - Exercises Set 1.6 - Page 106: 20

$$\frac{3}{4}$$

Given $$\lim_{x\to 0}\frac{\sin 6x}{\sin 8x}$$ Then \begin{align*} \lim_{x\to 0}\frac{\sin 6x}{\sin 8x}&=\lim_{x\to 0}\frac{\frac{\sin 6x}{6x}6x}{\frac{\sin 8x}{8x}8x}\\ &=\frac{6}{8}\lim_{x\to 0}\frac{\frac{\sin 6x}{6x} }{\frac{\sin 8x}{8x} } \\ &=\frac{6}{8}\frac{\lim_{x\to 0}\frac{\sin 6x}{6x} }{\lim_{x\to 0}\frac{\sin 8x}{8x} }\\ &=\frac{6}{8}\\ &=\frac{3}{4} \end{align*}