Evaluate the following limits \[\] $ \lim\limits_{ x \to 0} \large\frac{ \sin \: ax}{ \sin \: bx}, $$ a , b \neq 0$

Question

$\begin{array}{1 1}\large\frac{a}{b} \\ 0 \\ \large\frac{-a}{b} \\ \large\frac{b}{a}\end{array} $

Answer 1

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]$)