# Is there a relationship between the unbounded infinities and uncountable infinities? [duplicate]

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.

• Also, math.stackexchange.com/questions/linked/90758 has probably several other relevant threads. The short answer is "no", by the way. Commented Aug 13, 2015 at 15:28
• @AsafKaragila Sorry about that I just realized it should have been Beth-1 or the cardinal of the real numbers.Would that be a better question? Commented Aug 13, 2015 at 15:31
• It would be the same question. With the same "no" as an answer. And with the same temptation to mark this as a duplicate. Commented Aug 13, 2015 at 15:36
• @AsafKaragila Theres no need to mention that twice. Im just trying to satisfy curiosity. It sounds rude. Btw I flagged it a duplicate myself but if you want to also go ahead Commented Aug 13, 2015 at 15:44

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.