I came to understand that asymptotic negligibility, dominance and equivalence form together a partial order of real-valued functions, analogous to $=, \leqslant, <$ for the real numbers. In the following, I thus use $f \precsim g$ for $f=O(g)$ and $f \prec g$ for $f=o(g)$. We have:
$$ \begin{matrix} & \sim & \precsim & \prec \\ \text{transitivity} & \checkmark & \checkmark & \checkmark \\ \text{reflexivity} & \checkmark & \checkmark & \text{anti-} \\ \text{symetry} & \checkmark & \text{anti-} & \text{anti-}\end{matrix} $$
I like this analogy, because it gives me intuitions about things like: $f \sim g \implies (f \precsim g \text{ and } f \succsim g)$ or $(f \prec g \text{ and } g \precsim h) \implies f \prec h$.
Something boggles me though. Because in:
$$f \underset{c}\precsim g \iff \exists k \in \mathbb{R}, |f|< k |g| \text{ on some neighborhood of } c$$
... we define dominance for any $k$, we also don't have the (nice) property that: $$f \precsim g \text{ and } f \succsim g \implies f \sim g$$
For instance, I have both $x \precsim 3x+4$ and $x \succsim 3x+4$ but $x \nsim 3x+4$. I have always found the fact that $x \succsim 3x+4$ counter-intuitive! Actually, if we define dominance only for $k\geqslant 1$, it seems that the former property holds, since we now have $x \not\succsim 3x+4$.
My question is thus, is there any textbook that uses this definition of dominance (the one with $k\geqslant 1$) ? And if not, why not?