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A continuous function f(x) satisfies lim_(x -> 0) [x * ln(1 + x)] / [integral from 0 to x of f(t)*sin(t) dt] = 5. What is f(0)?

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

Correct answer: 2/5

Solution

As x -> 0, x*ln(1+x) ~ x². The integral from 0 to x of f(t)*sin(t)dt ~ f(0)*sin(0)*x... using the approximation sin(t) ~ t: integral ~ f(0)*x²/2. So the limit = x² / (f(0)*x²/2) = 2/f(0) = 5, giving f(0) = 2/5.

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