Question

Evaluate the given limit :
$\underset{x\to 0}{lim}\frac{\sin ax+bx}{ax+\sin bx},a,b,a+b\ne 0$

Answer 1

Given $\underset{x\to 0}{lim}\frac{\sin ax+bx}{ax+\sin bx}$
Substituting $x=0$ in the given limit,
$\underset{x\to 0}{lim}\frac{\sin ax+bx}{ax+\sin bx}=\underset{x\to 0}{lim}\frac{\sin a\left(0\right)+b\left(0\right)}{a\left(0\right)+\sin b\left(0\right)}$
$=\frac{0+0}{0+0}$
$=\frac{0}{0}$
Since it is in $\frac{0}{0}$ form.

We need to simplify it,
Let $L=\underset{x\to 0}{lim}\frac{\sin ax+bx}{ax+\sin bx}$
$L=\underset{x\to 0}{lim}\frac{x(\frac{\sin ax}{x}+b)}{x(a+\frac{\sin bx}{x})}$
$L=\underset{x\to 0}{lim}\frac{\left(\frac{\sin ax}{x}\right)+b}{a+\left(\frac{\sin bx}{x}\right)}$
Multiply and divide $\frac{\sin ax}{x}$ by $ax$ and multiply and divide $\frac{\sin bx}{x}$ by $bx$
$\Rightarrow L=\underset{x\to 0}{lim}\frac{(\frac{\sin ax}{x}\cdot \frac{ax}{ax})+b}{a+(\frac{\sin bx}{x}\cdot \frac{bx}{bx})}$
$\Rightarrow L=\underset{x\to 0}{lim}\frac{((\frac{\sin ax}{ax}\cdot \frac{ax}{x})+b)}{a+(\frac{\sin bx}{bx}\cdot \frac{bx}{x})}$
$\Rightarrow L=\underset{x\to 0}{lim}\frac{(\left(\frac{\sin ax}{ax}\right)\cdot a+b)}{a+\left(\frac{\sin bx}{bx}\right)b}$
$\Rightarrow L=\frac{\left(1\right)a+b}{a+\left(1\right)b}$
$\Rightarrow L=\frac{a+b}{a+b}$
$\Rightarrow L=1$