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Let functions $f(n)$ and $g(n)$ be increasing in $n$. I am trying to say the following precisely:

As $n\rightarrow\infty$, if $f(n)$ is "smaller" than $g(n)$ then $A$ is true, and if $f(n)$ is "the same or larger" than $g(n)$ then $B$ is true.

where, by $f(n)$ being "smaller" than $g(n)$ I mean that $\lim_{n\rightarrow\infty}g(n)-f(n)=\infty$, and by $f(n)$ being "the same" as $g(n)$ I mean $\lim_{n\rightarrow\infty}|g(n)-f(n)|=c<\infty$. Basically, here $g(n)$ "segments" (if there is such a term) the set of positive increasing real functions in two subsets. But, how do I state this compactly and precisely?

I thought of using the Landau notation in the following manner:

As $n\rightarrow\infty$, if $f(n)=o(g(n))$ then $A$ is true, and if $f(n)=\Omega(g(n))$ then $B$ is true.

Is this what I need? Does the union of the sets $o(g(n))\cup\Omega(g(n))$ contain the set of all positive increasing real functions and, if it does, then are the subsets the same as defined above? The definitions of $o(g(n))$ and $\Omega(g(n))$ involve limits of fractions, which makes me wary...

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