Comments by "MC116" (@angelmendez-rivera351) on "Find the Limit of sec(x) as x approaches negative pi/2 from the left" video.
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Let us stay with the real numbers before going into the complex numbers. Above, I explained that you can use the ordering of the real numbers to extend to the real numbers into a system where every subset has a supremum, and this introduces the objects T and B, which Wikipedia (and most non-mathematicians) call ∞ and –∞.
However, there is a different extension you can do with the real numbers. Rather than using the ordering, you can use a notion in the theory of topological spaces called a one-point compactification. The idea is similarly to that of the affinely extended real line, except that you take the two endpoints that this line has, and you join them together into one point. It is almost as if you are saying "let –∞ = ∞," but you can do this very rigorously, and it is actually very useful as well. While you do give up the ability to do arithmetic, like with the affinely extended real numbers, and while you also give up the ordering, unlike the affinely extended real numbers, it gives you one huge benefit in exchange: the ability to do projective geometry. As such, this structure is called the projectively extended real line. You can study Möbius transformations (whenever people write 1/0 = ∞, they are talking about a special case of an elementary Möbius transformation, they are not talking about division, and this is clearly another example of abuse of notation, which again, is by no means universal).
If you think about it visually, it amounts to turning the real line into a real "circle" of sorts, but a circle with an infinite circumference. This is intentional, because one grand motif in projective geometry is that lines are treated like circles, and parabolas are just treated like ellipses, and since circles are ellipses, they are all just ellipses in the context of projective geometry. Lines are just ellipses with infinite eccentricity, and parabolas are just ellipses with eccentricity 1. You can even get funky and treat hyperbolas like ellipses as well if you allow complex numbers. More importantly, this type of structure allows you to systematically study the different kinds of asymptotes that exist. One intuition behind this is that it makes calculus more symmetric, so to speak. Rather than speaking about approach T or B (∞ and –∞ respectively), you instead speak about approaching ∞ (again, we really should use a different symbol) from the right (the positive real numbers) or the left (the negative real numbers), which feels rather natural.
The projective extended complex plane, also known as the Riemann sphere, is actually just a trivial extension of the projectively extended real line: it is just the union of the complex numbers and the projectively extended real line. The Riemann sphere is extremely useful in complex analysis as it simplifies many different concepts, and it serves as the main inspiration of wheel theory. In your comment, you talked about the complex numbers having only one "infinity," and it refers precisely to this, the Riemann sphere.
This brings me back to why I said it is confusing to use the word "infinity" here. You said "the real numbers have 2 infinities, while the complex numbers have 1 infinity." This is clearly not the case, though. As you can see, there are two different ways you can extend the real numbers, one of them introducing two new objects (which you can think of as infinite if you want, but please do not call them "infinity"), and the other one introducing only object (this object completely unrelated to the T and B of the other extension). In principle, there are other rigorous ways you can extend the real numbers too to introduce infinite objects, though they may not necessarily be useful. As for the complex numbers, I only talked about the projective extension, but you absolutely can use an affine extension of the complex numbers if you want to, it is perfectly valid, just nowhere as useful as the Riemann sphere. This introduces not two new objects, but infinitely many new objects, actually. The intuition is that, in analogy to how the real line was closed off by two endpoints, the complex plane is being closed off by a border which, intuitively, is the shape of a circle with infinite circumference. Each new infinite object corresponds to a direction in the complex plane. You can come up with a toroidal extension, where the real axis and the imaginary axis both get projectively extended, but separately, so that you get two new different infinite objects: one imaginary, one real. You can do all sorts of other extensions, as long as they are mathematically coherent.
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To start with, you should probably stop using the word "infinity" to talk about these objects, because it is mostly nonsensical, and it does not correspond to the way the mathematical theory actually works, and so, you will likely just end up confusing yourself by trying to think of these objects all under the name "infinity." "Infinity" is not a mathematical object, it is just some vague, pseudo-mathematical concept. Yes, in mathematics, we work with infinite objects all the time, but these objects all have different names, they are not "infinity," and they are well-defined within a given context, but they are not denoted by the symbol ∞ that everyone uses. These objects all do different things, and what it means for them to be "infinite" means different things, depending on what they are. This is why, if you want to understand what is really happening, you should really stop thinking of "infinity" as more than just a concept, and instead, just at mathematical concepts for what they are. I believe this is the most useful advice I can give you to move forward with understanding these topics.
The mathematical object that we call the "real number system" is defined in such a way that it has two fundamental components. One of the components makes the real numbers form a field. This means you can add these objects, you can multiply them, and you can (mostly) divide them, and there are certain strict rules for how to do that. The other component is that of an ordering: you can order the real numbers, you can compare them. It makes sense to say that 0 is less than 1. It makes sense to say that 1/3 < 1/2. This is important, because not every mathematical structure has this. Now, the way the real numbers are ordered is actually very, very special. In what way is it special? In three ways: (a) addition and multiplication are compatible with the ordering; (b) the ordering is total, meaning that for any two real numbers x, y, you can always, without exception, compare them; (c) the ordering satisfies the least upper bound property. This last one may confuse you. What is the least upper bound property? It is the property that says, that if I take any arbitrary (non-empty) subset S of the real numbers, if S is bounded from above (it has an upper bound), then it must have a least upper bound. For example, consider the interval (0, 1) (the endpoints are excluded). This interval has an upper bound. 2 is an upper bound. 10 is an upper bound. π is an upper bound. However, of all the infinitely many upper bounds, one of them is the smallest one possible, and this is the real number 1. Why is this the smallest upper bound? Because any real number less than 1 is either in the interval (0, 1), or smaller than 0, and so, not an upper bound of (0, 1). Even though (0, 1) has no greatest real number, it does have a least upper bound, which is 1. By the way, the least upper bound is also called the supremum, and this is the name I will be using from now on.
This should make you think a little more about (c). The ordering is such that every nonempty set of real numbers bounded from above has a supremum, but it feels as though we should be able to make this even stronger. What happens if we extend the real numbers, in such a way that all sets of real numbers have a supremum? For this to be possible, there needs to exist some object greater than all real numbers, and this object will be the largest object in the set. I will call this object T, which stands for "top." There also needs to exist some object smaller than all real numbers, and this object will be the smallest in the set. This object needs to exist so that the empty set can have a supremum in this ordering. This object, I will call B, which stands for "bottom." Hence, we have the set of all real numbers, and also, the objects T and B. Together, this new ordered system is called the affinely extended real line, and the geometric, visual idea is that the line of real numbers has been extended in such a way, that it now has endpoints, B and T. If you read the Wikipedia article, or some other popular but non-scholarly source, then you will find that the symbols for T and B that they use are ∞ and –∞, and they usually are read "infinity" and "negative infinity." However, as I already explained to you, this notation/language is very confusing, and it is misleading, if not outright incoherent, and this is not universal in the mathematical literature either. So, although I do want you to be aware where the symbols –∞ and ∞ come from when doing calculus, I will not be using them, unless it amounts to clarifying something, because if I do use them, it will confuse you. Every time you see a statement of the form lim f(x) (x —> ∞) = L, you should replace this with lim f(x) (x —> T) = L, and similarly with –∞ and B. Also, every time you see lim f(x) (x —> p) = ∞, you should replace this with lim f(x) (x —> p) = T, and again, this analogous with –∞ and B.
One thing that is very important to understand is that, while you can do arithmetic with real numbers, you cannot do arithmetic with T and B (which is why people usually say "infinity is not a number"). Yes, these are valid, well-defined mathematical objects as far the ordering system is concerned, but you cannot perform addition, multiplication, or division with these objects, without creating a bunch of contradictions. This can be proven carefully, but I will not do that here. And I know that often, you will see these strange "conventions" where you see things like 1/∞ = 0 and x + ∞ = ∞, but these are just abuse of notation, and are not universal - different contexts use different conventions for abusing notation. Properly speaking, there actually is no such a thing as arithmetic with these objects. You can still do certain operations with these objects, like the supremum operation, of course, and you can still define many classes of functions for these objects whenever you are not relying on analysis rather than arithmetic, but these funcions and operations are not the operations we usually call "arithmetic." This is why it is very tricky and complicated to evaluate limits when it comes to expressions where these objects become involved in some way or another, and actually, you are not required to have these objects to do any calculus at all.
Anyway, this is all to say that the conceptual idea I just explained, of making sure all sets of real numbers have a supremum (an idea that is indeed very useful, as long as you are not trying to turn it into an arithmetic number system), is where the symbols –∞ and ∞ come from. Meanwhile, the usage of the symbol ∞ in the context of complex analysis, is completely, completely unrelated to this, and again, it just boils down to abuse of notation. But what is the actual underlying concept behind it? I will explain that in the next comment.
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