Question

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

Answer 1

$\underset{x\to 0}{lim}\frac{ax+x\cos x}{b\sin x}=\frac{1}{b}\underset{x\to 0}{lim}\frac{x(a+\cos x)}{\sin x}$
$=\frac{1}{b}\underset{x\to 0}{lim}\left(\frac{x}{\sin x}\right)\times \underset{x\to 0}{lim}(a+\cos x)$
$=\frac{1}{b}\times \frac{1}{\left({lim}_{x\to 0}\frac{\sin x}{x}\right)}\times \underset{x\to 0}{lim}(a+\cos x)$
= $\frac{1}{b}\times (a+\cos 0)=\frac{a+1}{b}$