It seems what you really want is a set of functions $A$ such that $$\forall (f,g\in A) f \leq g \Leftrightarrow \exists C\in\mathbb{R}:\lim_{x\rightarrow\infty}\frac{|f(x)|}{|g(x)|}\leq C$$ I think the set of functions that have non-zero finite limits qualifies. If you put in too many growth choices it fails because there is no $$lim_{x\rightarrow\infty}\frac{|f(x)|}{|g(x)|}$$ $\lim_{x\rightarrow\infty}\frac{|f(x)|}{|g(x)|}$$ for some pairs, if $g$ grows slower than $f$.
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It seems what you really want is a set of functions $A$ such that $$\forall (f,g\in A) f \leq g \Leftrightarrow \exists C\in\mathbb{R}:\lim_{x\rightarrow\infty}\frac{|f(x)|}{|g(x)|}\leq C$$ I think the set of functions that have non-zero finite limits qualifies. If you put in too many growth choices it fails because there is no $$lim_{x\rightarrow\infty}\frac{|f(x)|}{|g(x)|}$$ for some pairs, if $g$ grows slower than $f$. |
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