Question

Answer 1

$$\begin{gathered} \underset{x\to 0}{\lim }\left[\frac{ax+x\cos x}{b\sin x}\right] \\ \text{Dividing the numerator and the denominator by x:} \\ \underset{x\to 0}{\lim }\left[\frac{a+\cos x}{b\left({\displaystyle \frac{\sin x}{x}}\right)}\right]\left[\because \underset{x\to 0}{\lim }\frac{\sin x}{x}=1\right] \\ =\frac{a+\cos 0}{b} \\ =\frac{a+1}{b} \end{gathered}$$