A function $f:\mathbb R \to \mathbb R$ is slowly varying at infinity if for any $t>0$ $$ \lim_{x\to +\infty}\frac{f(xt)}{f(x)}=1. $$ Is there a bounded function slowly varying at infinity whose limit as $x \to +\infty$ does not exist?
2026-02-27 18:08:58.1772215738
Slowly varying function without limit at infinity
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How about $f(x)=2+\sin \ln \ln x$? Note that the difference between $\ln\ln(xt)$ and $\ln x$ tends to $0$ as $x\to\infty$.