logo

Ask Questions, Get Answers

X
 
Home  >>  CBSE XI  >>  Math  >>  Limits and Derivatives

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]$)
answered Apr 4, 2014 by thanvigandhi_1
edited May 16, 2014 by balaji.thirumalai
 

Related questions

...