Answer
$$\frac{a}{b }$$
Work Step by Step
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*}