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Evaluate the limit lim_(x→ 0)(1-cos³ x)/(xsin xcos x).

1. (2)/(5)
2. (3)/(5)
3. (3)/(2)
4. (3)/(4)

Correct answer: (3)/(2)

Solution

Near 0, 1-cos x ~ x^2/2 so 1-cos^3 x ~ 3(x^2/2), while x sin x cos x ~ x^2. The limit is (3/2).

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