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The following inequality is true for all x close to 0. 2 − x²/3 < x sin x/(1 − cos x) < 2 What is the value of lim(x→0) x sin x/(1 − cos x) ?

1. 0
2. 1/2
3. 1
4. 2

Correct answer: 2

Solution

As x approaches 0, the expression x sin x/(1 - cos x) simplifies to 2, which can be confirmed using L'Hôpital's rule or Taylor series expansion, showing that the limit converges to 2.

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