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If f(x) = { (sin(p+1)x + sin x)/x, x < 0; x/q, x = 0; (√(x + x²) − √x)/x^(3/2), x > 0 } is continuous at x = 0, then the ordered pair (p, q) is equal to:
- (-3/2, 1/2)
- (-1/2, 3/2)
- (-3/2, 1/2)
- (5/2, 1/2)
Correct answer: (-3/2, 1/2)
Solution
To ensure continuity at x = 0, the left-hand limit as x approaches 0 must equal the right-hand limit and the function value at x = 0. By calculating these limits and setting them equal, we find that p must be -3/2 and q must be 1/2 to satisfy the continuity condition.
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