I want to find a slowly varying function that has no limit at infinity, and write down its Karamata's representation .
That is, I want to find $L: (0,\infty) \to (0,\infty)$ with the $\lim_{x \to \infty} L(x)$ not existing, and I want to write down $L(x) = c(x) \exp(\int^x_1 \epsilon(t) t^{-1} dt$ where $c, \epsilon \colon (0,\infty) \to (0,\infty)$ with $c(x) \to c \in (0,\infty)$ as $x \to \infty$, and $\epsilon(t) \to 0$ as $t \to \infty$.
The definition I am using is from the book Extreme Values, Regular Variation and Point Processes.
I have read that $\sin \ln \ln x+2$ fits the bill, but I do not know its Karamata's representation.
I also thought
$L(x) = \exp(\sin \ln \ln x) = \exp(\int^x_e \frac{\cos \ln \ln t}{t \ln t} dt$) would work but $\frac{\cos \ln \ln t}{\ln t}$ is not strictly positive so it does not fit the requirement that $\epsilon(t)$ is strictly positive.
So my question is, what slowly varying function has no limit at infinity, and what is its Karamata's representation?