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Consider the function \[ f(x)=\begin{cases} \dfrac{1-\sin^3 x}{3\cos^2 x}, & x<\dfrac{\pi}{2},\\[4pt] p, & x=\dfrac{\pi}{2},\\[4pt] q(1-\sin x), & x>\dfrac{\pi}{2}. \end{cases} \] If \(f\) is continuous at \(x=\dfrac{\pi}{2}\), then the pair \((p,q)\) is

1. (1, 4)
2. (1/2, 2)
3. (1/2, 4)
4. None of these

Correct answer: (1/2, 2)

Solution

The function is continuous at x = π/2 if the limit from the left equals the limit from the right and both equal f(π/2). Evaluating the left limit gives 1/2 and the right limit gives 2, thus p must be 1/2 and q must be 2 for continuity.

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