Question

$\underset{x\to 0}{\lim }\frac{\tan \mathrm{x}-\sin x}{\sin 3x-3\sin x}$

Answer 1

$\underset{x\to 0}{\lim }\left[\frac{\tan \mathrm{x}-\sin x}{\sin 3x-3\sin x}\right]$

$$\begin{gathered} =\underset{x\to 0}{\lim }\left[\frac{{\displaystyle \frac{\mathrm{si}nx}{\cos x}}-\sin x}{3\sin x-4{\sin }^{3}x-3\sin x}\right] \\ =\underset{x\to 0}{\lim }\left[\frac{\sin x\left(1-\cos x\right)}{\cos x\times -4{\sin }^{3}x}\right] \\ =\underset{x\to 0}{\lim }\left[\frac{2{\sin }^2\left({\displaystyle \frac{x}{2}}\right)}{\cos x\times -4{\sin }^{2}x}\right] \\ =\underset{x\to 0}{\lim }\left[\frac{2\sin \left({\displaystyle \frac{x}{2}}\right)\times \sin \left({\displaystyle \frac{x}{2}}\right)}{\cos x\times \left(-4\sin x\right)\times \sin x}\right] \\ =\underset{x\to 0}{\lim }\left[\frac{2{\displaystyle \frac{\sin \left({\displaystyle \frac{x}{2}}\right)}{{\displaystyle \frac{x}{2}}}}\times {\displaystyle \frac{x}{2}}\times {\displaystyle \frac{\sin \left({\displaystyle \frac{x}{2}}\right)}{{\displaystyle \frac{x}{2}}}}\times {\displaystyle \frac{x}{2}}}{\cos x\times -4{\displaystyle \frac{\sin x}{x}}\times x\times {\displaystyle \frac{\sin x}{x}}\times x}\right] \\ =\underset{x\to 0}{\lim }\left[\frac{2\times {\displaystyle \frac{x}{2}}\times {\displaystyle \frac{x}{2}}}{\cos x\times -4\times x\times x}\right] \\ =\frac{-1}{8\cos 0} \\ =-\frac{1}{8} \end{gathered}$$