Let f:→R→(0,∞) be strictly increasing function such that limx→∞f(7x)f(x)=1. Then, the value of limx→∞[f(5x)f(x)−1] is equal to
Question
Let f:→R→(0,∞) be strictly increasing function such that limx→∞f(7x)f(x)=1. Then, the value of limx→∞[f(5x)f(x)−1] is equal to
Solution
The problem is asking for the limit of a function as x approaches infinity. Here are the steps to solve it:
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Given that lim(x→∞) [f(7x) / f(x)] = 1, we can infer that f(7x) ~ f(x) as x→∞. This is because the limit of the ratio of two functions is 1, which implies that the two functions are approximately equal for large x.
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We need to find the value of lim(x→∞) [f(5x) / f(x) - 1]. We can rewrite this as lim(x→∞) [f(5x) / f(x)] - lim(x→∞) 1.
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Since f(x) is a strictly increasing function, we can infer that f(5x) > f(x) for all x > 0. Therefore, the limit of the ratio f(5x) / f(x) as x approaches infinity should be greater than 1.
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However, we don't have enough information to determine the exact value of this limit. We know that it's greater than 1, but without more information about the function f(x), we can't find a specific numerical value.
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Therefore, the value of lim(x→∞) [f(5x) / f(x) - 1] is indeterminate based on the information given in the problem.
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