Question

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

Answer 1

Here $\underset{x\to 0}{\text{lim}}\frac{\text{sin ax+bx}}{\text{ax+sin bx}}\left[\frac{0}{0}\text{form}\right]$

Dividing numerator and denominator by ax,

$\underset{x\to 0}{\text{lim}}\frac{\frac{\text{sin ax}}{\text{ax}}+\frac{bx}{ax}}{\frac{ax}{ax}+\frac{\text{sin bx}}{ax}}=\underset{x\to 0}{\text{lim}}\frac{\frac{\text{sin ax}}{\text{ax}}+\frac{bx}{ax}}{\frac{ax}{ax}+\frac{\text{sin bx}}{bx}\times \frac{bx}{ax}}$

= $\frac{1+\frac{b}{a}}{1+\frac{b}{a}}$ = 1.