Question

Evaluate the given limit :
$\underset{x\to 0}{lim}\frac{\sin ax}{bx}$

Answer 1

Given : $\underset{x\to 0}{lim}\frac{\sin ax}{bx}$
Substituting $x=0$ in the given limit,
$\underset{x\to 0}{lim}\frac{\sin ax}{bx}=\frac{\sin a\left(0\right)}{b\times 0}$
$=\frac{\sin \left(0\right)}{0}$
$=\frac{0}{0}$
Since it is in $=\frac{0}{0}$ form,
We need to simplify it
$\Rightarrow \underset{x\to 0}{lim}\frac{\sin ax}{bx}$
$=\underset{x\to 0}{lim}\sin ax\times \frac{1}{bx}$
Multiplying and dividing by $ax$
$=\underset{x\to 0}{lim}\sin ax\times \frac{1}{bx}\times \frac{ax}{ax}$
$=\underset{x\to 0}{lim}\frac{\sin ax}{ax}\times \frac{ax}{bx}$
$=\underset{x\to 0}{lim}\frac{\sin ax}{ax}\times \frac{a}{b}(\because x\ne 0)$
$=\frac{a}{b}\times \underset{x\to 0}{lim}\frac{\sin ax}{ax}$
$\{\text{Using}\underset{x\to 0}{lim}\frac{\sin nx}{nx}=1\}$
$=\frac{a}{b}\times 1$
$=\frac{a}{b}$
$\therefore \underset{x\to 0}{lim}\frac{\sin ax}{bx}=\frac{a}{b}$