Question

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

Answer 1

$$\begin{gathered} \underset{x\to 0}{\lim }\left[\frac{\sin \left(ax\right)+bx}{ax+\sin \left(bx\right)}\right] \\ =\underset{x\to 0}{\lim }\left[\frac{{\displaystyle \frac{\sin ax}{ax}}\times ax+bx}{ax+{\displaystyle \frac{\sin bx}{bx}}\times bx}\right] \\ =\underset{x\to 0}{\lim }\left[\frac{\left({\displaystyle \frac{\sin ax}{ax}}\times a+b\right)x}{\left(a+{\displaystyle \frac{\sin bx}{bx}}\times b\right)x}\right] \\ =\frac{1\times a+b}{a+b} \\ =1 \end{gathered}$$