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Let f : R → R be a function defined as f(x) = { [sin(a+1)x + sin 2x]/2x , if x < 0; b/2 , if x = 0; [√(x + bx^3) − √x]/(bx^(5/2)) , if x > 0 } If f is continuous at x = 0, then the value of a + b is equal to:

1. −5/2
2. −2
3. −3
4. −3/2

Correct answer: −5/2

Solution

For the function to be continuous at x = 0, the limits from both sides as x approaches 0 must equal the value of the function at x = 0. By evaluating the left-hand limit and the right-hand limit, we find that both must equal b/2, leading to the equation that ultimately gives a + b = -5/2.

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