When a function f increases without bound we say $f(x)=\infty$. How does this idea relate to, if at all with the infinite sets we study in set theory?
To give a better understanding of why I'm bringing this up let me present and example,
Take $$\lim_{x\to2}f(x)$$ where $$f(x) = \frac{3}{(x-2)}$$ the graph of which is
Now obviously $\lim_{x\to2}f(x)$ does not exist but could we in some context say that $\lim_{x\to2}f(x)$ corresponds to(or is even equal or equivalent to) a set $\mathscr M$ such the |$\mathscr M$| = |$\mathscr{R}$|. In other words, if they are related then how is that and if not, why not? Thanks in advance.
I would say the relationship is weak at best.
There are two easy ways to append infinite values to the set of reals numbers: affinely, where we append $\{+\infty , -\infty\}$; or projectively, where we append only $\{\infty\}$. Both of these are just the real numbers with extra values that make the space compact, and make some statements a bit nicer. Its essentially a notational convenience for certain kinds of divergence or non real values.
This has barely anything to do with either the concept of cardinality or any particular cardinals. The infinities we append don't relate to the sizes of set, they are just convenient labels for certain function behaviours.