Answer
\[ = \frac{a}{b}\]
Work Step by Step
\[\begin{gathered}
\mathop {\lim }\limits_{x \to 0} \,\,\frac{{\sin ax}}{{\sin bx}} \hfill \\
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rewrite \hfill \\
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= \,\left( {\mathop {\lim }\limits_{x \to 0} \frac{{bx}}{{\sin bx}}} \right)\,\left( {\mathop {\lim }\limits_{x \to 0} \frac{{\sin ax}}{{ax}}} \right)\,\left( {\frac{a}{b}} \right) \hfill \\
\hfill \\
\frac{a}{b} = \,\left( {\mathop {\lim }\limits_{x \to 0} \frac{{bx}}{{\sin bx}}} \right)\,\left( {\mathop {\lim }\limits_{x \to 0} \frac{{\sin ax}}{{ax}}} \right)\, \hfill \\
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evaluate\,\,the\,\,special\,limits \hfill \\
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= \,\left( 1 \right)\,\left( 1 \right)\,\left( {\frac{a}{b}} \right) \hfill \\
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{\text{Simplify}} \hfill \\
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= \frac{a}{b} \hfill \\
\end{gathered} \]
