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Let f be a function on the interval (π/6, π/3) given by f(x) = { (√2 cos x − 1)/(cot x − 1), for x ≠ π/4; k, for x = π/4 } If f is continuous, what is the value of k?

1. 2
2. 1/2
3. 1
4. 1/√2

Correct answer: 1/2

Solution

Write f(x)=sin x*(sqrt2 cos x -1)/(cos x - sin x). With x=pi/4+h, sqrt2 cos x -1 ~ -h, cos x - sin x ~ -sqrt2 h, sin x ~ 1/sqrt2, so the limit is (1/sqrt2)(-h)/(-sqrt2 h)=1/2. Hence k=1/2.

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