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limit of f(tx)/f(x) as x->infinity, f has a constant sign

Source: VJIMC 2007 1.3

June 24, 2021
functionlimitsreal analysis

Problem Statement

A function f:[0,∞)→R∖{0}f:[0,\infty)\to\mathbb R\setminus\{0\} is called slowly changing if for any t>1t>1 the limit lim⁡x→∞f(tx)f(x)\lim_{x\to\infty}\frac{f(tx)}{f(x)} exists and is equal to 11. Is it true that every slowly changing function has for sufficiently large xx a constant sign (i.e., is it true that for every slowly changing ff there exists an NN such that for every x,y>Nx,y>N we have f(x)f(y)>0f(x)f(y)>0?)
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