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Consider the function f: (0,1) → R defined by f(x) = √n whenever x lies in the interval [1/(n+1), 1/n), where n belongs to the set of natural numbers. Let g: (0,1) → R be a function satisfying the inequality ∫x¹ (1−t)/t dt ≤ g(x) < 2√x for all x in (0,1). Determine the value of lim x→0 f(x)g(x).

1. The limit does not exist
2. The limit equals 1
3. The limit equals 2
4. The limit equals 3

Correct answer: The limit equals 2

Solution

The function f(x) is piecewise-defined, and g(x) is bounded by the given inequality. As x approaches 0, the behavior of both functions leads to their product approaching 2, based on the dominant terms in their respective expressions.

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