What's the difference between Complex infinity and undefined?

Question

Can somebody please expand upon the specific meaning of these two similar mathematical ideas and provide usage examples of each one? Thank you!

Answer 1

I don't think they are similar.

"Undefined" is something that one predicates of expressions. It means they don't refer to any mathematical object.

"Complex infinity", on the other hand, is itself a mathematical object. It's a point in the space $\mathbb C\cup\{\infty\}$, and there is such a thing as an open neighborhood of that point, as with any other point. One can say of a rational function, for example $(2x-3)/(x+5)$, that its value at $\infty$ is $2$ and its value at $-5$ is $\infty$.

To say that $f(z)$ approaches $\infty$ as $z$ approaches $a$, means that for any $R>0$, there exists $\delta>0$ such that $|f(z)|>R$ whenever $0<|z-a|<\delta$. That's one of the major ways in which $\infty$ comes up. An expression like $\lim\limits_{z\to\infty}\cos z$ is undefined; the limit doesn't exist.