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If lim (x -> 2) of [x^(5/2) - x^(3/2) + x² - x - 2*x^(1/2) - 2] / (x - 2) = 3 * (sqrt(P) + 1), find the value of P where P is a natural number.

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

Correct answer: 2

Solution

The limit is f'(2) where f(x) = x^(5/2) - x^(3/2) + x² - x - 2*sqrt(x) - 2. f'(x) = (5/2)*x^(3/2) - (3/2)*x^(1/2) + 2x - 1 - 1/sqrt(x). At x=2: f'(2) = (5/2)*(2*sqrt(2)) - (3/2)*sqrt(2) + 4 - 1 - 1/sqrt(2) = 5*sqrt(2) - (3/2)*sqrt(2) + 3 - 1/sqrt(2). Converting to common form: (5 - 3/2 - 1/2)*sqrt(2) + 3 = 3*sqrt(2) + 3 = 3*(sqrt(2)+1) = 3*(sqrt(P)+1). So sqrt(P) = sqrt(2) => P = 2.

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