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Consider f(x) = lim_(n -> infinity) ((1 + cos x)ⁿ + 5 * ln(x)) / (2 + (1 + cos x)ⁿ). Which of the following is true?

1. f(x) is continuous at positive odd multiples of pi
2. f(x) is discontinuous at positive even multiples of pi
3. f(x) is discontinuous at positive odd multiples of pi/2
4. f(x) is continuous at positive even multiples of pi/2

Correct answer: f(x) is continuous at positive odd multiples of pi

Solution

Let u = 1 + cos x. Then f(x) = lim (uⁿ + 5 ln x)/(2 + uⁿ). Case 1: 0 < u < 1 (i.e., -1 < cos x < 0, i.e., x in (pi/2, pi) mod 2pi): uⁿ -> 0. f(x) = (0 + 5ln x)/(2 + 0) = 5 ln x / 2. Case 2: u = 0 (x = (2k-1)*pi, odd multiples of pi): uⁿ = 0. f(x) = 5 ln x / 2. Case 3: u = 1 (x = (2k-1)*pi/2 + pi = odd multiples of pi/2 not equal to pi): 1ⁿ = 1. f(x) = (1 + 5ln x)/3. Case 4: 1 < u < 2 (i.e., 0 < cos x < 1, i.e., x near 0 or 2pi): uⁿ -> inf. f(x) = lim (uⁿ + 5ln x)/(2 + uⁿ) -> 1. Case 5: u = 2 (x = 2k*pi, even multiples of pi): uⁿ -> inf. f(x) -> 1. At x = (2k-1)*pi (odd multiples of pi): f = 5 ln x / 2 (from case 2). Approaching from left (u near 0, case 1): f -> 5 ln x / 2. Approaching from right (u near 0, case 1): f -> 5 ln x / 2. So f is continuous at odd multiples of pi. Option A is TRUE.

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