This definition uses hyper-reals and nonstandard analysis.
Let $k^*(x)$ be the natural extension of $k(x)$. Let $f$ and $g$ be functions.
$f = O(g) := \frac {f^*(H)} {g^*(H)}$ is finite for all positive unlimited $H$.
$f=\Omega(g) := \frac {g^*(H)} {f^*(H)}$ is finite for all positive unlimited $H$.
$f=\Theta(g):=f=O(g) \wedge f=\Omega(g)$
$f=o(g):=\frac {f^*(H)} {g^*(H)}$ is infinitesimal for all positive unlimited $H$.
$f=\omega(g):=\frac {g^*(H)} {f^*(H)}$ is infinitesimal for all positive unlimited $H$.
I am not familiar with the Omega- and Theta- notations, but the big-O and little-o definitions are correct. There is a bit of an ambiguity in your notation since you didn't specify where $x$ tends to exactly, but I assume $x$ is increasing without bound.
To respond to robjohn's question: as far as comparison between the traditional and the hyperreal approaches, the little-o and big-O example is not particularly enlightening. A better example is the definition of continuity of a function $f$. You are probably aware of the fact that a majority of undergraduates never fully master the traditional $\epsilon,\delta$ definition of continuity. It may come as more of a surprise that Cauchy never defined continuity using $\epsilon,\delta$. In fact, Cauchy defined continuity in terms of the property that an infinitesimal $x$-increment necessarily produces an infinitesimal change in $y=f(x)$. This is what the hyperreal definition says, as well.