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A piecewise function f(x) is defined as: f(x) = (1 - sin³(x)) / (3*cos²(x)) for x < pi/2; f(x) = a for x = pi/2; f(x) = b*(1 - sin(x)) / (pi - 2x)² for x > pi/2. If f is continuous at x = pi/2, find the values of a and b.

1. a = 1/2, b = 1/4
2. a = 2, b = 4
3. a = 1/2, b = 4
4. a = 1/4, b = 2

Correct answer: a = 1/2, b = 4

Solution

Left limit (x -> pi/2⁻): Let h = pi/2 - x -> 0+. sin(x) = cos(h), cos(x) = sin(h). Numerator: 1 - cos³(h) = (1-cos(h))(1+cos(h)+cos²(h)). Denominator: 3*sin²(h). As h->0: (1-cos(h))/sin²(h) = (1-cos(h))/(1-cos²(h)) * (1-cos(h))... use (1-cos(h))/h² -> 1/2 and sin²(h)/h² -> 1. So LHL = (h²/2)*(1+1+1)/(3h²) = 3/(2*3) = 1/2. So a = 1/2. Right limit (x -> pi/2⁺): Let h = x - pi/2 -> 0+. sin(x) = cos(h), pi - 2x = pi - 2(pi/2+h) = -2h. (1-sin(x)) = (1-cos(h)) ~ h²/2. (pi-2x)² = 4h². RHL = b*(h²/2)/(4h²) = b/8. For continuity: b/8 = 1/2 => b = 4.

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