Chapter 1 - Limits and Continuity - 1.6 Continuity of Trigonometric Functions - Exercises Set 1.6 - Page 106: 19

$$\frac{7}{3}$$

Given $$\lim_{x\to 0}\frac{\tan 7x}{\sin 3x}$$ Then \begin{align*} \lim_{x\to 0}\frac{\tan 3x}{\sin 3x} &= \lim_{x\to 0}\frac{\frac{\tan 7x}{7x}7x}{\frac{\sin 3x}{3x}3x} \\ &=\frac{7\lim_{7x\to 0}\frac{\tan 7x}{7x}}{3\lim_{3x\to 0}\frac{\sin 3x}{3x}}\\ &=\frac{7}{3} \end{align*}