Chapter 1 - Limits and Continuity - 1.6 Continuity of Trigonometric Functions - Exercises Set 1.6 - Page 106: 31

$$\frac{a}{b }$$

Given $$\lim_{x\to 0} \frac{\tan ax }{\sin bx},\ \ a,b\neq0 $$ Then \begin{align*} \lim_{x\to 0} \frac{\tan ax }{\sin bx}&= \frac{\lim_{x\to 0}\frac{\tan ax}{ax}ax }{\lim_{x\to 0}\frac{\sin bx}{bx}bx}\\ &= \frac{\lim_{x\to 0}\frac{\tan ax}{ax}a }{\lim_{x\to 0}\frac{\sin bx}{bx}b}\\ &=\frac{a}{b } \end{align*}