Tangent-minus-sine third-order limit

Evaluate $\lim_{x\to 0} \dfrac{\tan x - \sin x}{x^3}$ . (to 3 decimals)

Show hints (2)+

Factor $\sin x$ out of the numerator: $\tan x - \sin x = \sin x\,\tfrac{1-\cos x}{\cos x}$ .
Then use $\tfrac{\sin x}{x}\to1$ and $\tfrac{1-\cos x}{x^2}\to\tfrac12$ .

Answer

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0.5 (± 0.001)

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Asked at: Probability & Market-Making, Timed Mental-Math & Sequences