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For the piecewise-defined function f(x) = { [sin((a+1)x) + sin x]/x, x < 0; c, x = 0; [√(x + bx²) - √x]/(b x^(3/2)), x > 0 } to be continuous at x = 0, which values of a, b, and c are required?

1. a = -3/2, c = 1/2, b = 0
2. a = 3/2, c = 1/2, b ≠ 0
3. a = -3/2, c = 1/2, b ≠ 0
4. None of these

Correct answer: a = -3/2, c = 1/2, b ≠ 0

Solution

The function must be continuous at x = 0, which requires that the limits from both sides equal the value at x = 0. The chosen values of a, b, and c ensure that the left-hand limit approaches 1/2 and the right-hand limit also approaches 1/2 as x approaches 0, thus satisfying the continuity condition.

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