# Evaluate the following limits  $\lim\limits_{ x \to 0} \large\frac{ \sin \: ax}{ \sin \: bx}, $$a , b \neq 0 \begin{array}{1 1}\large\frac{a}{b} \\ 0 \\ \large\frac{-a}{b} \\ \large\frac{b}{a}\end{array} ## 1 Answer Toolbox: • \bigg[ \lim\limits_{y \to 0} \large\frac{\sin \: y}{y}$$=1 \bigg]$
At $x = 0$ the value of the given function is $\large\frac{0}{0}.$
Using $\large\frac{ \sin \: ax}{ \sin \: bx}$$= \large\frac{ \bigg( \Large\frac{\sin\: ax}{ax} \bigg) \normalsize \times ax}{\bigg( \Large\frac{\sin\: bx}{bx} \bigg) \normalsize \times bx} = \bigg( \large\frac{a}{b} \bigg)$$ \times \large\frac{ \bigg( \Large\frac{\sin \: ax}{ax} \bigg)}{ \bigg( \Large\frac{\sin \: bx}{bx} \bigg)}$
$\Rightarrow$ $\lim\limits_{ x \to 0} \large\frac{ \sin \: ax}{ \sin \: bx}, $$a , b \neq 0 = \bigg( \large\frac{a}{b} \bigg)$$ \times \large\frac{ \lim\limits_{ax \to 0} \bigg( \Large\frac{\sin \: ax}{ax} \bigg)}{ \lim\limits_{bx \to 0} \bigg( \Large\frac{\sin \: bx}{bx} \bigg)}$
Since $x \to 0 \Rightarrow ax \to 0$ and $x \to 0 \Rightarrow bx \to 0$,
$\Rightarrow$$\lim\limits_{ x \to 0} \large\frac{ \sin \: ax}{ \sin \: bx},$$ a , b \neq 0$$= \bigg( \large\frac{a}{b} \bigg) (Using:\bigg[ \lim\limits_{y \to 0} \large\frac{\sin \: y}{y}$$=1 \bigg]$)
edited May 16, 2014