There exist a relationship between the properties $i)$ and $ii)$? Let $f:\mathbb{R}\to\mathbb{R}$ be a continuos function. Consider the properties:
$$i) \liminf_{|x|\to\infty} xf(x) =c>0$$
and
$$ii)\lim_{|x|\to\infty}\frac{f(x)}{e^{a|x|^b}}=0,\quad\forall a>0 \quad (b>0);$$
I am trying to understand if there is a relation between $i)$ and $ii)$, I mean if $i)\implies ii)$ and/or $ii)\implies i)$.
About me there is no relationship between $i)$ and $ii)$ but I can't find counterexamples so far.
Could someone please help me?
Thank you in advance.
 A: Counterexample, $(ii)$ does not imply $(i)$:

Let $f(x)=\ln(1+x^2)$.

Counterexample, $(i)$ does not imply $(ii)$ (there’s a risk this is piecewise continuous since I slightly botched together all the elements I needed, but it works, and the principle stands: someone more experienced than myself in mollifying could probably patch this more smoothly):

Fix any $b\gt0$. Define $f(x)=e^{\sin(\pi x)x^{b+1}}$ on the interval $[0,1]$. For any $n\in\Bbb N$, put $f(x)$ on the interval $[n,n+1/2]$ as $f(x)=\frac{1}{n}e^{\sin(\pi(x-n))\cdot x^{b+1}}$, and on $[n+1/2,n+1]$ as $f(x)=\frac{g(x)}{n+1}e^{\sin(\pi(n+1-x))\cdot x^{b+1}}$, where $g(x)=-\frac{2}{n}x+3+\frac{2}{n}$ on $[n+1/2,n]$. $f$ is continuous and extends similarly to a definition  way to a definition for negative $x$. You find that the limit inferior of $x\cdot f(x)$ is $1=c\gt0$ yet the limit $(ii)$ does not exist, since it blows up to infinity near $|x|=n+1/2$ and vanishes near $|x|=n$.

Motivation: for the limit $i)$ to exist, we need to have a sequence of points at which $f$ behaves like $1/x$, yet inbetween the points of this sequence $f$ needs to grow extremely rapidly in order for limit $ii)$ not to exist. $g$ interpolates between the $1/n$ points so as to make the function continuous.
There should be a workable link to the graph here (ignore the output for negative $x$).
