A race between two small-angle rates

Evaluate $\displaystyle\lim_{x\to 0}\frac{1-\cos 2x}{x\,\sin 3x}$ . Give a decimal to four places.

Show hints (2)+

Use $1-\cos 2x \approx 2x^2$ (mind the $2x$ squaring to $4x^2$ ) and $\sin 3x\approx 3x$ .
Both top and bottom are order $x^2$ ; the limit is the ratio of coefficients $\tfrac{2}{3}$ .

Answer

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0.6667 (± 0.001)

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