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Let p, q, r be positive real numbers with p + q + r = 21. The limit lim(x -> 0) ((p^x + q^x + r^x) / 3)^(1/x) cannot be equal to which of the following?

1. 7
2. 4
3. 8
4. 5

Correct answer: 8

Solution

The limit equals the geometric mean (pqr)^(1/3). By AM-GM, the geometric mean is at most 7 (the arithmetic mean). So any value greater than 7 is impossible, making 8 the answer.

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