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Let h(x) = h(2x) for every real x, where h is a continuous function and h(2012) = pi/2. Define the limit M = lim_(x->0) [cos²(h(x)) + 1 - sin³(h(x))] / sin²(x). What is the value of 4M?

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

Correct answer: 0

Solution

The functional equation h(x) = h(2x) implies h(x) = h(x/2ⁿ) for all n; letting n -> infinity and using continuity gives h(x) = h(0) = pi/2 for all x. The numerator becomes cos²(pi/2) + 1 - sin³(pi/2) = 0 + 1 - 1 = 0, identically in x, so M = 0 and 4M = 0.

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