TITLE: Slowly varying function without limit at infinity
QUESTION [5 upvotes]: 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?

REPLY [7 votes]: How about $f(x)=2+\sin \ln \ln x$? Note that the difference between $\ln\ln(xt)$ and $\ln x$ tends to $0$ as $x\to\infty$.