 It kind of makes sense in some cases to actually talk about what would an arithmetic calculation look like if it involved infinity, right? Like if you take the limit as x approaches infinity of 2 plus x. Well, if we were just to naively plug in the number infinity, this would look like 2 times infinity, right? But like we talked about a moment ago with polynomials, the leading term is all that matters. So this should also be the same thing as x approaches infinity of just x, which would be infinity as we learned with polynomials before. And so it makes sense that we should say that 2 plus infinity is equal to infinity, right? We can make like an arithmetic statement about these type of things. Now, I mentioned this is very dark magic, right? Because whenever we make a statement about an arithmetic statement with infinity like r plus infinity, r divided by infinity, infinity times infinity, these are really just shorthand statements for limit calculations. So one has to tread very lightly right now. And so if I was to liken these calculus calculations with chemistry, then we really are now verging on alchemy at the moment. And so doing arithmetic with infinity is essentially the calculus equivalent of human transmutation, one that's to be very careful that we don't blow our bodies up or something like that, something horrific, right? Horrible things have happened to necromancers. That's not necromancers, excuse me, with alchemists who have played around with the laws of equivalent exchange and things like that. So we really are becoming full-metal numerologists at this moment. So be very cautious with these things. Now, these first statements I'm going to present in front of you, these are actually fairly safe statements. You can use these without any real concern whatsoever. If you ever take a real number r, so in all these cases, r is just any real number. It can be positive. It can be negative. It can be zero. I don't care. If you take any real number and you add it to infinity, you can always simplify that calculation as infinity itself. r plus infinity will equal infinity. Likewise, if you take a real number and you add it to negative infinity, that will just give you negative infinity. And this is sort of like if I take the ocean and add a bucket of water, it's still the ocean. If I take the ocean and I try to take away a bucket of water, it doesn't make a difference. So infinity compared to a finite, the infinity always wins. Also, if you take infinity plus infinity, if you take two oceans and combine them together, that's still an ocean. Same thing with negative infinity plus negative infinity is equal to negative infinity. So none of those statements are in disagreement with each other. An infinite plus a finite will be infinite. And if you have compatible infinity, so those equal that same infinity. Now with that said, I mentioned, right, we have to make sure our summoning circles are drawn correctly here. Otherwise we might summon the wrong demon, right? One of these demons in numerology here, we refer to as indeterminate forms. There are certain forms involving infinity or zero, which is sort of like the reciprocal infinity we'll see in just a moment. There are some calculus forms for which we do not have enough information to determine what the limit's gonna be. And therefore I can't give you a general statement of this arithmetic with infinity. And the first one we run across is if you ever have infinity minus infinity, which is the same thing saying like plus infinity plus negative infinity or negative infinity plus infinity, something like that. If you have to add together an infinite and a negative infinite, then it turns out that's an indeterminate form. Let me give you such an example, right? If you take the limit as x approaches infinity of x cubed minus x squared, this looks like infinity minus infinity. If you were just to plug in naively, you're gonna get infinity cubed minus infinity squared, which will become infinity minus infinity. We'll get to that in just a second. The power's infinity. That looks like infinity minus infinity. But like we said, because of the leading term being x cubed, it should just be the limit of x cubed as x approaches infinity, which would be, of course, in that case, infinity, right? So we see that in this case, infinity minus infinity should be infinity. But if we were to switch things up a little bit, if I took negative x cubed plus x, this actually would look like negative x cubed plus, sorry, negative infinity cubed plus infinity. You get negative infinity plus infinity again. So that's the same thing. But as you have a negative x cubed as the leading term, the outcome should be negative infinity. We can see that infinity minus infinity could be positive or negative infinity. It turns out it could also be a finite number as well. As a very simple case, if I took negative x cubed plus x cubed, that's gonna look like infinity minus infinity. But in that case, it should be zero because negative x cubed plus x cubed can simplify to be zero. But it turns out in other bizarre cases, I can get seven, right? Infinity minus infinity might be seven. If you have an infinite army versus an infinite army and they battle each other to the doom, it might turn out there's seven survivors of that conflict. Who knows? It depends on the function. We'll explore more of that in the future. That's why you need to be cautious of arithmetic of infinity. If you're very not careful, it turns out you might blow off an arm or something. I hate all these graphic imagery here, but you need to be equally cautious when you work with infinity, right? Working with infinity is kind of like division by zero. You might blow up yourself if you're not careful. Now, for the rest of these ones, consider when C is itself a positive real number. It's any real number that's positive, right? So if you take C times infinity, so a positive times positive, that should be positive and it'll be positive infinity. If you have a double negative, right? So if C is positive, negative C is negative. If you take a negative times negative infinity, that's also gonna be positive infinity. You saw something like that before, right? Particularly when we had negative two times infinity, we said that was negative infinity or a positive times negative infinity, that should be negative infinity as well. So when it comes to multiplication with signs, when you throw infinity in that case, it has the same rules as before. Positive times a positive is positive. A double negative is positive. A positive times a negative is negative. A negative times a positive is negative, like so. If you have infinity times infinity, that's infinity. That would be like an infinity squared. If you have negative infinity times negative infinity, that would be, that's positive infinity, right? It's a double negative. And so that would also allow us to do things like infinity cubed, infinity to the fourth. Infinity cubed just means infinity times infinity times infinity, which you can use recursion on that to calculate such statements. So multiplication with infinities is fairly straightforward. It mostly just needs to pay attention to the sign. Division gets a little bit more complicated, right? But there are some situations where things are quite simple. If you have the expression r divided by infinity, if you take an infinite number and divide that from a finite number, that's gonna give you zero. And that's even true if the numerator was zero. That's okay. Notice I used an r right here. That zero is okay. If you take zero divided by infinity, notice there's no conflict here because if you times by zero, the product wants to be zero. But if you divide by infinity, it wants the quotient to be zero. There's no disagreement there. They both wanna be zero, so you're gonna get zero right there. That's gonna lead to some other things we'll see in just a moment, right? Now you might wanna be a little bit more cautious when it comes to zero, right? Cause maybe are you approaching zero from above or below? That depends again on the sign. If you take like c over infinity, that's gonna look like zero plus. Because if you take a positive divided by positive, that'll be positive. And as that absolute value gets smaller, smaller, smaller close to zero, it's gonna approach zero from above. If you take negative c over negative infinity, same thing, that the double negatives will cancel if you get a positive. On the other hand, if you take like negative c divided by infinity, that's gonna give you zero from below. If you take like positive c divided by negative infinity, that also give you zero from below. So if we do need to pay attention to the sign like we did with the horizontal asymptote a moment ago, then by all means, we can pay attention to zero plus or zero minus. Same thing going on right here. If you take our divided by negative infinity, that'll be zero, but we can distinguish between positive and negative cases by looking at the signs. So this is what happens when you divide by infinity here. But there are two important exceptions when it comes to division. There's the case where you get zero divided by zero and you get the case of infinity divided by infinity. So when you multiply by zero, like I already mentioned, multiplication by zero is dominant. That is when you multiply by zero, the outcome wants to be zero. But on the other hand, when you divide by zero, that wants to be infinite. When we talked about vertical asymptotes previously, if you take a real number divided by zero, where the real number's non-zero itself, when you take our divided by zero, that wants to be infinite. Paws are negative depending on if we're approaching zero from the right or zero from the left right there. And so division by zero is itself not an indeterminate form. It's not a real number, but if you take a number divided by zero that typically gives you plus or minus infinity, that's not indeterminate. But if you divide by zero, it wants to be infinite. But if you times by zero, it wants to be zero. So which one is it? Much like zero minus zero, you get this like clash of titans, these two monstrous beasts that want to destroy each other. And we're afraid that grease will be destroyed in the process, right? Who's the more dominant term, multiplication by zero or division by zero? It turns out it depends on the specific functions in play. Zero divided by zero is an indeterminate form. And we have seen many cases of this already in this lecture series, where division by zero divided by zero turned out to be a finite number. Think of the limits of difference quotients we've considered already. Those oftentimes, the data forms zero divided by zero, but often turn out to be a finite number. We've seen somewhere it was zero, we've seen somewhere it was infinite. Infinity divided by infinity is basically the same thing. If you times by infinity, it wants to be infinity. If you divide by infinity, it wants to be zero. There's this competition, this clash of titans. Who's the stronger titan? It depends on the functions that produced this infinity in the first place. It turns out infinity divided by infinity is essentially the same thing as zero divided by zero. Because in some respect, dividing by infinity gives you zero. It's kind of like zero and infinity are reciprocals of each other. But this is not exactly true, that we can't really divide by zero. But if we kind of in this extended real number system, they kind of don't work. And we see that zero divided by zero and infinity divided by infinity are kind of the same thing. Another indeterminate form you have to be very cautious about is zero times infinity, right? Notice how I took a positive number times infinity, or I took a negative number times infinity. That's okay. If you take zero times infinity, there's a clash of titans there. Multiplication by zero wants to give you zero. Multiplication by infinity wants to give you infinity. So which one is the more dominant term? And it turns out it depends on the functions that produce zero and infinity there. It could be zero, it could be infinity, it could be one, it could be anything. And so those clash of titans you have to watch out for. There are three other exponential indeterminate forms we also have to watch out for. These ones are a little bit harder to believe, but I promise you there's also this clash of titans. If you take zero to the zero power, the clash there is powers of zero want to be zero. But if you raise something to the zero power, you want to get one. So which one is it, one or zero? In the case of a conflict, in case of this clash of titans, it turns out anything is possible. Same thing is also true for infinity to the zero. Well, if you take powers of infinity, that should be infinity, but something to the zero power should be one. So which one is it? In the presence of conflict, it turns out it could be anything. We have to investigate the functions more thoroughly. And then the last one, this one students very much disagree with often, but it's the same thing applies. If you take one to the infinity, one to the infinity, that is an indeterminate form. Because yeah, you're like, well, all powers of one are one, you're right. But when you raise something to an infinite power, that should be infinity. There's a clash of titans there, isn't there? So one to infinity is not necessarily one. I can give you such an example. If we take the limit as x approaches infinity, and we take the expression one plus one over n to the n power. If we take that expression right there, you'll notice this thing looks like one to the infinity, right? Because in a slightly more detail, this would look like one plus one over infinity to the infinity. By the arithmetic we've seen before, you get one over infinity, which is zero, one plus zero should be one. So this looks like one to the infinity, but the actual limit here is the number e, that natural exponential we've seen before. So we have to be very cautious with this arithmetic at infinity. These calculations are justifiable, but only if you can avoid these indeterminate forms. When you hit an indeterminate form, you're gonna have to approach it differently.