With x measured in radians, evaluate the limit of (tan 2x − 2x)/(x − sin x) as x goes to 0. The answer is an integer.

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Tangent against sine

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With $x$ measured in radians, evaluate the limit of $\frac{tan 2 x - 2 x}{x - sin x}$ as $x$ goes to $0$ . The answer is an integer.

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Hint

The numerator and the denominator vanish to the same order; compare their leading terms.

Solution

From the series $tan u = u + \frac{u ^{3}}{3} + \dots$ with $u = 2 x$ , we get $tan 2 x - 2 x = \frac{8 x ^{3}}{3} + \dots$ .
Similarly $sin x = x - \frac{x ^{3}}{6} + \dots$ , so $x - sin x = \frac{x ^{3}}{6} + \dots$ .
The ratio of the leading terms is $\frac{\frac{8}{3}}{\frac{1}{6}} = 16$ .
So the limit equals $16$ .

Answer: $16$