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Let f: R → R be a differentiable function with f(1) = 4. If f''(1) = 2, then evaluate \nlim_{x→1} \frac{\int_{4}^{x} 2t\,dt}{x-1}.

1. 16
2. 8
3. 4
4. 2

Correct answer: 4

Solution

The limit evaluates the average value of the integral of the function 2t from 4 to x as x approaches 1. By applying the Fundamental Theorem of Calculus and L'Hôpital's Rule, we find that the limit simplifies to 4, which corresponds to the value of the function at the point of interest.

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