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The values of p and q for which the function f(x) = { [sin(p + 1)x + sin x]/x, x < 0; q, x = 0; [√(x + x²) - √x]/x^(3/2), x > 0 } is continuous for all x in R, are
- p = 5/2, q = 1/2
- p = -3/2, q = 1/2
- p = 1/2, q = 3/2
- p = 1/2, q = -3/2
Correct answer: p = -3/2, q = 1/2
Solution
As x->0-, [sin(p+1)x+sin x]/x -> (p+1)+1 = p+2. As x->0+, [sqrt(x+x^2)-sqrt(x)]/x^(3/2) = (sqrt(1+x)-1)/x -> 1/2, so q=1/2. Continuity needs p+2=1/2, giving p=-3/2, q=1/2.
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