Problem 12 - Olympiad

Let f(x) = { (x^2 - 4) / (x - 2) if x < 2; ax + b if x ≥ 2 }. For what values of a and b is f continuous at x = 2?

A. a = 1, b = 1B. a = 4, b = -4C. a = 2, b = 3D. a = 3, b = 0

Correct: B

For x < 2, f(x) = (x^2 - 4) / (x - 2) = (x - 2)(x + 2) / (x - 2) = x + 2. So lim_(x→2⁻) f(x) = 2 + 2 = 4. For continuity at x = 2, we need lim_(x→2⁺) f(x) = f(2) = 4. For x ≥ 2, f(x) = ax + b, so lim_(x→2⁺) f(x) = 2a + b. Setting 2a + b = 4: For choice A: 2(1) + 1 = 3 ≠ 4. For choice B: 2(4) + (-4) = 8 - 4 = 4 ✓. For choice C: 2(2) + 3 = 4 + 3 = 7 ≠ 4. For choice D: 2(3) + 0 = 6 ≠ 4. Therefore a = 4 and b = -4 makes f continuous at x = 2.