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If f(x) is continuous and satisfies lim(x->0) [x * ln(1+x)] / [integral from 0 to x of f(t)*sin(t) dt] = 5, find f(0).

1. 2/5
2. 10
3. 5
4. 5/2

Correct answer: 2/5

Solution

As x -> 0: numerator x*ln(1+x) ~ x². Denominator integral₀^x f(t)sin(t)dt ~ f(0)*x²/2 (since sin t ~ t near 0). Limit = x²/(f(0)*x²/2) = 2/f(0) = 5, so f(0) = 2/5.

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