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Question
\[\lim_{x \to 0} \frac{\tan x - \sin x}{\sin 3x - 3 \sin x}\]
Solution
\[\lim_{x \to 0} \left[ \frac{\tan x - \sin x}{\sin 3x - 3 \sin x} \right]\]
\[ = \lim_{x \to 0} \left[ \frac{\sin x\left( 1 - \cos x \right)}{\cos x \times - 4 \sin^3 x} \right]\]
\[ = \lim_{x \to 0} \left[ \frac{2 \sin^2 \left( \frac{x}{2} \right)}{\cos x \times - 4 \sin^2 x} \right]\]
\[ = \lim_{x \to 0} \left[ \frac{2\sin\left( \frac{x}{2} \right) \times \sin\left( \frac{x}{2} \right)}{\cos x \times \left( - 4 \sin x \right) \times \sin x} \right]\]
\[ = \lim_{x \to 0} \left[ \frac{2\frac{\sin\left( \frac{x}{2} \right)}{\frac{x}{2}} \times \frac{x}{2} \times \frac{\sin\left( \frac{x}{2} \right)}{\frac{x}{2}} \times \frac{x}{2}}{\cos x \times - 4\frac{\sin x}{x} \times x \times \frac{\sin x}{x} \times x} \right]\]
\[ = \lim_{x \to 0} \left[ \frac{2 \times \frac{x}{2} \times \frac{x}{2}}{\cos x \times - 4 \times x \times x} \right]\]
\[ = \frac{- 1}{8 \cos0}\]
\[ = - \frac{1}{8}\]
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