An infinite limit exists or doesn't exist?

Question

I have several months dealing with the situation of the existence of a limit. The way I teach to my students is the limit at a exists if the two-sided limits exist and are the same.

Most calculus textbooks say that limits as Lim x→0 (1/x^2)=∞ but the limit doesn't exist. However, they introduce the extended definition of a limit. Stewart Essential Calculus, page 56, states, "This does not mean that we regard ∞ as a number. Nor does it mean that the limit exists. It simply expresses the particular way in which the limit does not exist".

I understand this point of view is when a limit exists, it should be a finite number otherwise; it doesn't exist. It sounds weird to me because I understand if the limit is infinity, then the function increases without upper (or lower) bound.

My question is. Isn't this confusing for the student that Lim x→0 (1/x^2)=∞ doesn't exist? I would say the limit exists and is infinity. And the situation where the limit doesn't exist is when the two-sided limits are not the same or when the function is oscillating (as sin(1/x) as x→0).

I discussed this issue with some of my colleagues, and I concluded the answer to this issue is the topology where we are working. In calculus, one works with the standard topology (-∞,∞) and when the limit is infinity, the function approaches to a point out from the space, so it makes sense to say Lim x→0 (1/x^2)=∞ doesn't exist. But if one works with the extended topology [-∞,∞] then Lim x→0 (1/x^2)=∞ exists and is infinity.

In any of the above situations, the infinity is a cluster point of the function beyond the existence or not of the limit. In the end, I think it is a matter of pure convention. But I prefer to say the limit exists.

I direct my question to both tutors and students. I want to know your opinions about this topic which has been of long discussion along years.

Thank you for your time in reading this!

Best,

Tomás

Answer 1

Hi Thomas,

Your question is a bit of a technical one. If we consider the real numbers, R, then it doesn't make sense to say that the limit is equal to infinity. The line lim_x-->a f(x)=infinity is just meaningless. Recall the epsilon definition of a limit existing.

Conversely, we CAN make sense of a limit existing if it's infinity by considering the extended real line- also called the two point compactification of the reals. Here, it makes complete sense to label infinity as a "point" and then you are free to say that the limit is indeed infinity.

As this is calculus one and there's no notion of a topology for students at that level, we typically just tell them that the limit does not exist in order to avoid these technical details.

Best,

Andrew