Let $ f: R \rightarrow R $ be a positive increasing function with $ \lim _{x \rightarrow \infty} \frac{f(3 x)}{f(x)}=1 $. Then $ \lim _{x \rightarrow \infty} \frac{f(2 x)}{f(x)}= $

Question

Let $ f: R \rightarrow R $ be a positive increasing function with $ \lim _{x \rightarrow \infty} \frac{f(3 x)}{f(x)}=1 $. Then $ \lim _{x \rightarrow \infty} \frac{f(2 x)}{f(x)}= $, $ 1 $, $ \frac{2}{3} $, $ \frac{3}{2} $, $ 3 $

MARKS App · Accepted Answer

As f is a positive increasing function, we have f x f 2 x f 3 x Dividing by f x leads to 1 f ⁡ 2 x f ⁡ x f ⁡ 3 x f ⁡ x As lim x → ∞ f ⁡ 3 x f ⁡ x = 1 , we have by squeeze theorem or sandwich theorem, lim x → ∞ f ⁡ 2 x f ⁡ x = 1

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