# A Linear Order from AP Calculus

In teaching my calculus students about limits and function domination, we ran into the class of functions

$$\Theta=\{x^\alpha (\ln{x})^\beta\}_{(\alpha,\beta)\in\mathbb{R}^2}$$

Suppose we say that $g$ weakly dominates $f$, and write $f\preceq g$, if

$$\lim_{x\to\infty}\frac{f(x)}{g(x)} \hspace{3 mm} \text{is finite}$$

We can then readily see that $(\Theta,\preceq)$ is a total order isomorphic to the lexicographic order on $\mathbb{R}^2$.

But we can get more complicated total orders with, say

$$\Theta_n=(x^{\alpha_0}(\ln{x})^{\alpha_1}(\ln\ln{x})^{\alpha_2}\cdots(\ln^{n-1} x)^{\alpha_{n-1}})_{\vec{\alpha}\in\mathbb{R}^{n}}$$

$$\Phi=\{e^{p(x)}\}_{p(x)\in\mathbb{R}[x]}$$

which are isomorphic as total orders to the lexicographic orders on $\mathbb{R}^n$ and $\operatorname{List}\mathbb{R}$

All of these complicated orders live inside what I'd call "the AP Calc linear order" $(\Omega,\preceq)$ defined as:

$$\Omega_0=\{f\in\mathscr{C}^0((\lambda,\infty))\}_{\lambda\in\mathbb{R}}$$

$$f\preceq g\Longleftrightarrow \max\left\{\left|\liminf_{x\to\infty} \frac{f(x)}{g(x)}\right|,\left|\limsup_{x\to\infty} \frac{f(x)}{g(x)}\right|\right\}<\infty$$

$$\Omega=\Omega_0/\simeq \hspace{5 mm} \text{where} \hspace{5 mm} f\simeq g \Leftrightarrow \left[f\preceq g \text{ and } g\preceq f\right]$$

where the refinement on $\preceq$ is made so as to avoid problems with things like $\sin{x}$.

This seems to be a very complicated linear order, as it includes as a suborder things like

$$\Psi=\{p_0(x)e^{p_1(x)}e^{e^{p_2(x)}}\cdots\exp^{n-1}(p_{n-1}(x))\}_{p_i(x)\in\mathbb{R}[x]\forall i}$$

My question is the following: is there any combinatorial description or universal construction, i.e. as a colimit, of the isomorphism type of $(\Omega,\preceq)$?

• – David E Speyer Aug 27 '15 at 16:19
• This is not a total order; $f$ and $g$ can oscillate between $f\gg g$ and $g\gg f$. – Eric Wofsey Aug 27 '15 at 17:49
• A side comment: The class $\Theta$ is a very small sub-class of the logarithmico-exponential functions studied by Hardy (see also Boshernitzan's work). These have very nice properties including no oscillation: for any two distinct functions in a Hardy Field, eventually $f$ is bigger than $g$ or vice versa. – Anthony Quas Aug 27 '15 at 18:34
• @DmitryV: That's not what I was talking about (I was also thinking like a category theorist and not caring about that). If $f/g$ oscillates between approaching $0$ and approaching $\infty$, then $f\not\leq g$ and $g\not\leq f$. Actually, if $f$ and $g$ don't have to be positive, you could just have an unbounded set where $f$ is zero and $g$ is not and an unbounded set where $g$ is zero and $f$ is not. – Eric Wofsey Aug 27 '15 at 23:39
• With Anthony Quas, I suggest you look up "Hardy field", and also "o-minimal structure" whose germs at infinity produce Hardy fields. The question as it stands suffers from problems noted by Eric Wofsey. – Todd Trimble Aug 27 '15 at 23:47