Question

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

Answer 1

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

$\underset{x\to 0}{\text{lim}}\frac{\text{ax+x cos x}}{\text{b sin x}}=[\frac{\text{ax}}{\text{b sin x}}+\frac{\text{x cos x}}{\text{b sin x}}]$

= $\underset{x\to 0}{\text{lim}}[\frac{a}{b}\times \frac{\frac{1}{\text{sin x}}}{\text{x}}+\frac{\frac{1}{\text{b sin x}}}{\text{x cos x}}]$

= $\underset{x\to 0}{\text{lim}}[\frac{a}{b}\times \frac{\frac{1}{\text{sin x}}}{\text{x}}+\frac{\frac{1}{\text{b tan x}}}{\text{x}}]$

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