 In this video, I want to explore calculating some limits of various functions. We're going to put a lot of emphasis on piecewise functions in this video because piecewise functions can actually produce some pretty interesting questions about limits. Now, when you look at the graph, if I were to pick something like this right here, let's think of this right here. If I take my a value to be one, if I start getting closer and closer to one, either from the left or the right, the graph is going to start approaching, approaching, approaching, approaching. You get this point right here, which seems to be a little bit above y equals one. It turns out the function right here is behaving the way you would expect it. What I mean by that is that the function evaluation, this function called an H, this function H of one is actually equal to its limit as X approaches one of H of X here. This is something we'll talk about a little bit later on in chapter two about continuity. The function is that the function evaluation behaves the same way as the expectation that is F of one equals the limit as X approaches one. And when you think of most points in the domain of this function, that's exactly what you would expect. Like if I come in equal to X equals two as well, we see that the graph is so to speak continuous right here, that as you get closer and closer to X equals two, the behavior of the graph is that what you would expect it to be. When you get closer to X equals two, the Y coordinate is going to be closer and closer to two. All right. What I want to talk about is sort of what happens when you don't get what you expect, right? Because again, we could go day after day talking about the expectation here, right? If I were to take the limit here as X approaches zero of H of X here, we can see that as you approach zero from the right, it's gonna get you Y equals one. As you approach zero from the left, you're gonna get Y equals one. And therefore we would expect the limit to the function we expect to be one and the limit is the expectation. So the limit's gonna equal one because that's how the function is behaving. What happens if we come over to negative two, right? This is sort of an interesting creature right here. If we come over to negative two, you'll notice the following. If I approach negative two from the left, you're gonna end up at this point up here at the Y coordinate of four. And so what this tells us is that the limit as X approaches negative two from the left of H of X here, we would expect the function to be four. And when I say expect, I mean we can see nothing. We see nothing to the right of negative two. We can't see anything. If we can only see what's happening from the left, then it's like, hey, it should be four, all right? But it goes the other way around. Whoops, let me put that back on the screen. I'll just erase this. If we go the other way around, if we wanna figure out what is the limit as X approaches negative two from the right of H of X, you see in this situation, again, totally ignoring what happens up here. Like if we can't see it, if we can only see the things that are to the right of X equals negative two, we're gonna get closer and closer and closer to this point right here, which looks like Y equals two, right? Clean that up a little bit. The limit from the right is gonna be two. The limit from the left is gonna be one. And so what we then have to say in conclusion is that because the left limit is four and the right limit is two, the limit itself is actually undefined, right? The limit as X approaches negative two of H of X right here, it does not exist, right? The thing is, imagine we're like a judge, right? We have to decide a jury, right? We have to decide the outcome of some incident that happened, some crime, like did this person do it or not? Well, we have one evidence that's like four is guilty and then the other one's like two is guilty. It's like, well, which one is it, right? Is it four or two? The evidence is conflicting. Therefore, the limit turns out to not exist, right? So because of this, you see this actually type of jump on it, like the function or this case, it's a drop. Because the function has a huge drop on it, the left limit and the right limit don't agree with each other, which says the limit doesn't exist here. Another limit I want to consider is what if we take the limit as X approaches three of H of X, okay? So what would happen in this situation if we focus on X equals three for a moment, as we get closer and closer and closer to X equals three from the left, we're gonna get closer and closer and closer to this value right here. And so that's something we can definitely say. It's like, okay, the limit as X approaches three from the left of H of X, the function seems to be suggesting we get this value right here, which I drew that line poorly, but that's like, that's a three, right? We'll say that's close enough. So the limit as X approaches three from the left of H of X is gonna be three. But what happens when we approach it from the right? Well, when you approach it from the right, you don't see anything. There's nothing here to approach from, right? The closest you can is if you're over here, you're like, hey, what's happening over there? I can't see you too far away. You can't get close to it. And so what happens this time, what happens this time is as you approach three from the right, there is no such approach. So we actually can't describe the approach from the right. But this actually turns out to mean that the limit as X approaches three is also gonna equal three right here because there's no conflicting report. There's no conflicting evidence witnesses here. This, what we have right here is one witness says, if you approach three, you're gonna get three. The other witness didn't show up to trial. Therefore, it's like, well, okay, all the evidence says that H is gonna become three, okay? Jury, make your vote. The verdict says the limit's three. If there is no right-hand side to approach from, that means the limit will just be the left-handed limit. Similar statements can be said over here about this point. You'll notice that you can approach X equals five from the right. And so definitely the limit as X approaches five from the right here of H of X, this is gonna be zero. But there is no approach from the left. Therefore, because there's no conflicting opinions, then this tells us that the limit as X approaches five of H of X, this is also gonna be zero. So this happens when you have these end points. You only have to approach it from one side because there's only one side because it's on the end of the domain there. But you can also get some interesting things here like this jump we saw, that the left limit and the right limit disagreed with each other. Therefore, the limit didn't exist. Let's look at another example. When you look at this graph, this is the graph of the function, the absolute value of X over X, which I want you to think about that function for a second, the absolute value of X over X. If we treat this as a piecewise function, right? If X is greater than one, excuse me, greater than zero, if it's a positive number, in that situation, if you're positive, you're gonna get X over X, which actually simplifies to be one. And so the function is just gonna equal one. On the other hand, if you take X to be less than zero, well, if you have a negative, the absolute value will make it positive. So you actually, you're gonna get negative X on top, which is the opposite sign. And then you get X on bottom, it simplifies to be negative one. So you get this function, it's kind of a fun little function. This is the function that's one when you're positive, it's negative when you're negative. It's negative one when you're negative, right? So you see that right here. So it looks like a flat line y equals one when you're positive. And it looks like a flat line y equals negative one when you're negative. Notice that it is undefined at zero, right? If you plug in X equals zero, you get division by zero. And so this function, we'd be interested in what's the limit as X approaches zero. Well, notice that if we approach zero from the left, if you take the limit as X approaches zero from the left, the absolute value of X over X, you end up with negative one. The function predicts negative one if you get really, really close to zero from the left. On the other side, if you get closer and closer and closer to zero from the right, then you're gonna get the limit as zero approaches, as X approaches zero from the right, absolute value of X over X. In this case, they expect the outcome to be one, right? It looks like, it's like, hey, if you're asked someone over here, it's like, oh, I see a one. But then over here, if you're like, oh, this person's standing upside down, their world has been mirror-imaged. If they look at it like, oh, I see a negative one. They disagree with each other. And because of the conflicting evidence here, we then have to say that, oh, the limit doesn't exist. So we would then say that the limit as X approaches zero here does not exist. Because the left limit and the right limit don't agree with each other. They have to agree with each other in order to have the limit exist. And of course, if one of them didn't exist like we saw in the previous example, because there's nothing there to approach from, there's no disagreement there. There's no argument. Consider this graph right here. Take the limit as X approaches two of G of X, where G of X equals X squared plus four over X minus two. If we approach X equals two from the right, we're gonna be going up over here, over here, over here. We're gonna end up with the limit as X approaches two from the right of G of X right here. Well, this one, as you get closer and closer to the so-called vertical asymptote here, this would just keep on going, going, going, going, going, going, going, going. It wants to go to infinity and beyond. Actually, no, it just wants to go to infinity. Unlike Buzz Lightyear, it's not as ambitious. It just gets to infinity and we stop there. As X approaches two from the right, G will approach infinity, okay? We sometimes might abbreviate this as like, as X approaches two from the right, we would get that G of X approaches infinity, although this notation's a little bit more clear on what's going on there. What about the other side, okay? If I start approaching two from the left, I'm not going off towards infinity. I'm going off towards negative infinity. So as X approaches two from the left of G of X here, we end up with negative infinity. And the functions don't agree with each other then, for which then one of them's like, oh, I'm going towards infinity. The other one says, I'm going towards negative infinity. And so what is the function to a, what should we assume about the function? It's like, well, you're going to infinity. You're going to negative infinity. So what's G actually doing? Well, the limit as X approaches two of G of X, this does not exist because it can't be both positive and negative infinity. Those are two different quantities. Here's another example with a vertical acetote, although the, I think the ending is a lot more happy if you want the limit to be well-defined here. So in this situation, if we ask, what's the limit as X approaches zero of one over X squared? Well, if you approach from the left over here, the limit as X approaches zero from the left, one over X squared. This says, oh, we're going to approach infinity. But if you approach from the right, you'll bigger, bigger, bigger, bigger, bigger, bigger, you're going to get the limit as X approaches zero from the right of one over X squared. This likewise is going to be infinity. And so there's no disagreement. There's no discord right now. The left wants to be infinity. The right wants to be infinity. And if I dare make a political joke right here, it's like, oh, this is a bipartisan thing. The left and the right agree with each other. And therefore we can then say the limit as X approaches zero of one over X squared would equal infinity as well.