Evaluate the Given limit:

$\lim _{x \rightarrow 0} \frac{a x+x \cos x}{b \sin x}$

At $x=0$, the value of the given function takes the form $\frac{0}{0}$

Now,

$\lim _{x \rightarrow 0} \frac{a x+x \cos x}{b \sin x}=\frac{1}{b} \lim _{x \rightarrow 0} \frac{x(a+\cos x)}{\sin x}$

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

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

$=\frac{1}{b} \times(a+\cos 0) \quad\left[\lim _{x \rightarrow 0} \frac{\sin x}{x}=1\right]$

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