Question

$\underset{x\to 0}{\lim }\frac{5x+4\sin 3x}{4\sin 2x+7x}$

Answer 1

$$\begin{gathered} \underset{x\to 0}{\lim }\left[\frac{5x+4\sin 3x}{4\sin 2x+7x}\right] \\ =\underset{x\to 0}{\lim }\left[\frac{5x+4\times {\displaystyle \frac{\sin 3x}{3x}}\times 3x}{{\displaystyle \frac{4\sin 2x}{2x}}\times 2x+7x}\right] \\ =\underset{x\to 0}{\lim }\left[\frac{\left(5+4{\displaystyle \frac{\sin 3x\times 3}{3x}}\right)x}{\left(4{\displaystyle \frac{\sin 2x}{2x}}\times 2+7\right)x}\right] \\ =\frac{5+4\times 3}{4\times 2+7}\left[\because \underset{x\to 0}{\lim }\frac{\sin \left(3x\right)}{3x}=1\right] \\ =\frac{17}{15} \end{gathered}$$