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Let f: R → R be a positive, increasing function such that lim_(x→∞) (f(3x))/(f(x)) = 1. Then what is lim_(x→∞) (f(2x))/(f(x))?

1. 2/3
2. 3/2
3. 3
4. 1

Correct answer: 1

Solution

The limit condition indicates that as x approaches infinity, the growth rate of f(3x) and f(x) becomes similar, suggesting that f(x) grows at a rate that is asymptotically linear. Therefore, by the same reasoning, the limit of f(2x) compared to f(x) will also approach 1, indicating that f(2x) and f(x) grow at the same rate as x approaches infinity.

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