Question

$\underset{x\to 0}{\lim }\frac{\sin 3x+7x}{4x+\sin 2x}$

Answer 1

$$\begin{gathered} \underset{x\to 0}{\lim }\left[\frac{\sin 3x+7x}{4x+\sin 2x}\right] \\ =\underset{x\to 0}{\lim }\left[\frac{{\displaystyle \frac{\sin 3x}{3x}}\times 3x+7x}{4x+{\displaystyle \frac{\sin 2x}{2x}}\times 2x}\right] \\ =\underset{x\to 0}{\lim }\left[\frac{\left({\displaystyle \frac{\sin 3x}{3x}}\times 3+7\right)x}{\left(4+{\displaystyle \frac{\sin 2x}{2x}}\times 2\right)x}\right] \\ =\frac{3+7}{4+2} \\ =\frac{10}{6}=\frac{5}{3} \end{gathered}$$