Question 69

Let $$f: \mathbb{R} \rightarrow (0, \infty)$$ be strictly increasing function such that $$\lim_{x \to \infty} \frac{f(7x)}{f(x)} = 1$$. Then, the value of $$\lim_{x \to \infty} \left[\frac{f(5x)}{f(x)} - 1\right]$$ is equal to

f: R→(0,∞) strictly increasing, lim f(7x)/f(x)=1 as x→∞. Since f is increasing and this limit is 1, f grows sub-polynomially.

Since f(x)≤f(5x)≤f(7x), dividing by f(x): 1≤f(5x)/f(x)≤f(7x)/f(x)→1. So f(5x)/f(x)→1.

lim[f(5x)/f(x)-1]=0.

The answer is Option (2): 0.

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