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If lim_(x->0) [(1 + sin(x/2))(1 + sin(x/2²))(1 + sin(x/2³))]^(1/x) = e^(p/q), where p and q are relatively prime integers, find q - p.

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

Correct answer: 1

Solution

Taking logarithm: ln L = lim_(x->0) (1/x) [ln(1+sin(x/2)) + ln(1+sin(x/2²)) + ln(1+sin(x/2³))]. For small x, ln(1+sin(u)) ~ sin(u) ~ u. So ln L = 1/2 + 1/4 + 1/8 = 7/8. Thus L = e^(7/8), giving p=7, q=8, and q-p = 1.

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