Question

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

Answer 1

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

$$=\underset{x\to 0}{lim}\left[\frac{\frac{sinx}{\cos x}-\sin x}{3\sin x-4{\sin }^{3}x-3\sin x}\right]$$
$$=\underset{x\to 0}{lim}\left[\frac{\sin x(1-\cos x)}{\cos x\times -4{\sin }^{3}x}\right]$$
$$=\underset{x\to 0}{lim}\left[\frac{2{\sin }^2\left(\frac{x}{2}\right)}{\cos x\times -4{\sin }^{2}x}\right]$$
$$=\underset{x\to 0}{lim}\left[\frac{2\sin \left(\frac{x}{2}\right)\times \sin \left(\frac{x}{2}\right)}{\cos x\times (-4\sin x)\times \sin x}\right]$$
$$=\underset{x\to 0}{lim}\left[\frac{2\frac{\sin \left(\frac{x}{2}\right)}{\frac{x}{2}}\times \frac{x}{2}\times \frac{\sin \left(\frac{x}{2}\right)}{\frac{x}{2}}\times \frac{x}{2}}{\cos x\times -4\frac{\sin x}{x}\times x\times \frac{\sin x}{x}\times x}\right]$$
$$=\underset{x\to 0}{lim}\left[\frac{2\times \frac{x}{2}\times \frac{x}{2}}{\cos x\times -4\times x\times x}\right]$$
$$=\frac{-1}{8\cos 0}$$
$$=-\frac{1}{8}$$