 Welcome back to our lecture series Math 1210, Calculus I for students at Southern Utah University. As usual, I'll be your professor today, Dr. Andrew Misseldine. In this first video for lecture nine, we're going to talk about the idea of a limit some more, which you will recall that in our previous video, we talked about the precise definition of a limit. Let's just say that that led to a little bit of difficulty. We had talked about this idea of error and allowance that if we wanted to be sufficiently close to a Y-coordinate, how much error should we allow in the X-coordinate to guarantee things like that? This led to what we called the precise definition of a limit, but which it was difficult to really understand what was going on there. We did prove the limit of a line was such and such, but it was very difficult. Let's in this video introduce more of an intuitive notion of what a limit is, and we'll try to draw connections between these two notions. I think this will help simplify a lot of the concerns we might have seen previously. Imagine we have a function f of x equals x squared minus 4 over x minus 2. This is a standard rational function, and we might be asked to evaluate what is f of 1. If we were just to directly substitute in the value x equals 1 into this function, what would we get? Well, by direct substitution, f of 1, you're going to place each of these x's with a 1. In which case, you're going to get in the numerator, a 1 squared minus 4. 1 squared is 1, 1 minus 4 is a negative 3. In the denominator, you're going to get 1 minus 2, which is a negative 1, and see this double negative right here, negative 3 divided by negative 1 equals a positive 3. The function is 3 when x equals 1, so y equals 3 when x equals 1. This is the type of algebraic calculation that we do all the time. Now, why this is relevant to calculus is the next statement here. Let me draw it to your attention that it's also true that when x is a number very close to 1, then f of x is a number very close to 3. That is, if we accept a small amount of error above or below y equals 3, then there's a small amount of allowance to the left or right of x equals 1, which keeps y equals f of x inside of the margin of error. You notice I squiggled and was screeching with my voice, why did I do that for very close? Well, when we say something close, this is a relative term. If you are sitting in a classroom and someone was sitting close to you, that probably means they're like a few inches away from you. It's like, hey, Bob, I can smell you. Please, please scoot down a little bit. But if you're an asteroid speeding through our solar system, if you're within like 10,000 miles of Earth, that's considered really close, right? Closeness is somewhat relative, right? And so how do we make sense on these relatives terms in any precise sense? How can I say really close when it depends on what type of astronomical scale are we on? Are we talking about close inside the galaxy? We're separated by a few light years. Are we close on like the quantum scale? Oh yeah, these things are like one nanometer away from each other. Or what scale is going on here? So the definition, the precise definition of the limit allows for us to be precise on what we by very close. It means if I prescribe for you some type of Epsilon, so Epsilon is greater than zero. So there's some positive amount. Epsilon, we wanna think of as a really, really, really small number, right? Again, small is relative, right? But it's really, really small. So if we're just Epsilon away from three, that's what we consider very close. And so in correspondence, what does it mean to be very close to one? It means that we have some Delta greater than zero, right? There's some number Delta, so that if we're Delta close to one, then we'll be Epsilon close to three. So if we are close, Delta close to one, then F will always be Epsilon close to three, where of course we can derive Delta given any Epsilon whatsoever, okay? So intuitively, we can understand what things like close actually mean. But from a quantitative sense, we do need to be more precise because otherwise the ambiguity, we can get stuck in logical traps and things like that. That's the need for this Epsilon Delta stuff. All right, so why do we care about things being close, right? F of one equals three. Why does it matter if one, when you're close to one, the function will be close to three? Well, there are settings where we can't evaluate the function, but we can still talk about closeness. Let's take the same function and transition now to the number X equals two. You'll notice if you take your function F of X equals X squared minus four over X minus two, what happens when you plug in two, right? You're gonna get F of two. You don't see the doom that's approaching you. You get two squared minus four over two minus two. The danger's coming. It's like in that horror film when you can hear the music, then that the person's like, don't go in that room. What happens? You end up with getting zero over zero. We just divide by zero. We blew up our universe. This function is undefined at X equals two. F of two does not exist. D and E happening right here. So we can't evaluate the function at two, but we can evaluate the function near two. Because after all, if I look at F of 2.1, that's inside the domain. 2.1 doesn't make the denominator go to zero. Two does. I could calculate it at 2.01. I could calculate it 2.001. I could calculate it 2.000, lots of zeros, one at the end, right? All of those are gonna be well-defined because none of those make the denominator go to zero. That's the only problem with the domain. So I can get really close to two, right? Never touching, but I can get as close as two as you want. And so the proverbial family road trip, right? What happens, at least this was the case when I was a kid. We didn't have cell phones and Nintendo switches and things that we could bring in the car with us. We had to stare out of the cars as we're driving and look at a license plate. That's what kids didn't entertain themselves, you know? So what happened, of course, is the little brother, AKA me, I found enjoyment by pestering my older siblings, my older sister, right? So like, how close, how close can I get to my sister with my finger without actually touching her? And then, you know, she'll wind mom, Andrew's trying to touch me. It's like, I'm not touching you. I'm not touching you. I might be epsilon close, but I'm not touching you, right? We're not actually touching X equals two, but we can get really, really close to it. So it turns out that if we get really close, never actually equaling, but if we get really close to two, the function's still defined. And what happens to the function? As we get really, really close to two, it turns out the function will get really close to Y equals four. So even though F of two is undefined, there is a number that kind of suggests what it ought to be. What could F of two be if it lived up to its potential? It could be four. Look at this graph that we see right here. So you'll notice that when I'm saying things like, oh, you're gonna get really close to X equals one. Never actually touching it, but if I get numbers that get really, really, really close to X equals one, you'll notice coming up here and coming up here, right? If you get numbers really close to X equals one, your Y coordinates will get really close to three. So as X gets close to one, F of X will get close to three, the Y coordinate, never actually using, I don't actually have to use the number one there. But what if we do the same thing for X equals two? As my numbers get really close to X equals two, like I'm getting numbers super close on the number line to X equals two, then on the Y axis, your coordinates are gonna get really close to four. Now on the graph, you're gonna notice this notation here that when you see this solid dot, it's sometimes called a closed dot. That means the point is included on the graph. One comma three is a point on the graph. F of one equals three. On the other hand, you sometimes see this notation right here. Sort of like this empty, not filled in dot, they sometimes call it an open dot. This open dot indicates that while our function gets close, our graph gets close to two comma four, it's not actually included on the graph. Two comma four is not a point on the graph. F of two is undefined, it's not four, it's not 17, it's not 5,280, F of two does not exist, but the graph gets really, really, really close to the point two comma four. In fact, if I were to pick any acceptable amount of error, you could make this strip right here as small as I want. I could pick any epsilon greater than zero, and that could be a really small number. I could take epsilon to be one trillionth or even smaller. Ant-Man can shrink as small as he wants, he can get trapped in the quantum zone, and I could still pick an epsilon smaller than that. And if I did that, there would always be, there would always be some delta that I could find such that as long as I'm inside of the blue strip, then I'll land inside the yellow strip when I'm done. So I can focus in, I can zoom in on the point two comma four as much as I want, and it's never gonna fall outside of this box, no matter how small it is, as long as I'm near x equals two. So even though the function is undefined at x equals two, I can still look at the behavior of the graph near x equals two, and that's what the idea of the limit was all about. So we say things like the limit of F of x as x approaches two is equal to four or in shorthand notation we say this, the limit as x approaches two of F of x is equal to four. So as x gets close to two, F of x will get close to four. That doesn't mean the function is defined at two, it doesn't mean F of two equals four, but it means the behavior of the function will be close to four when you get close to two. It's also true that as x approaches one, F of x will get close to three. So no matter how close to three you want, you can always get close enough to one to guarantee that thing to happen. And so this is what we mean by this intuitive notion of a limit. Let's kind of revisit the definition, but now in this intuitive notion, right? So we say things like x approaches a from the left. So how you wanna interpret something like this is, if you have a number line and you have some specific number right here, so maybe this is like our x equals a value right here. If we talk about approaching x from, sorry, if x approaches a from the left, that means we're taking all these points on the number line that are getting closer and closer and closer and closer to x equals a, never actually touching a. And we will denote this as x arrow a, and you put a superscript of a negative one. That just means approaching it from the left. Cause after all, if you're to the left of the y-axis, that's the negative side. If you're to the right of the y-axis, that's the positive side. So a superscript negative sign means you approach from the left. Similarly, we can talk about as x approaches a from the right. What this means is you're talking about a sequence of numbers that's approaching a from the right, never touching a, but getting as close to it as you want, closer than the brother trying to poke his sister without actually poking her, right? And so we denote this as x approaches a plus. That means you approach it from the right. And so with these shorthands, we can talk about the limit from the left or the left-handed limit or the limit from the right. So the limit, the left-handed limit of a function is the limit as you approach x from the left will be denoted the limit as x approaches a from the left. You see it right there of f of x. So this is saying if I were to get delta close to a on the left, what should I expect the function to do? So as you arbitrarily get closer and closer to x equals a, what happens to the y-coordinate? If you don't see the right-hand side, you can't see it, you can't see it. If you only see the left-hand side of the graph what's the function behaving like? And similarly, you get the idea of a limit from the right. This is so-called right-handed limit. As x approaches a from the right, what is your function doing? That's called the right-handed limit. Now, if we allow approaching x equals a from both sides, this is the standard limit, sometimes called the two-handed limit, the double-sided limit to emphasize it's both left and right. But if you just see me write x arrow a, that means as you approach x approaches a, which means you approach x approaches a from the left and from the right, this gives you the standard notion of the limit, the limit as x approaches a of f of x, like here. And so let's look at another example. This one's kind of similar to what we saw a moment ago. If I were to pick any number in the domain, let's take x equals zero, right? As I get closer and closer to x equals zero, your y-coordinates are getting closer and closer to y equals one. And so then we would say something like the limit as x approaches zero for our function g of x right here, this is gonna be one. If I pick the value, let's say, let's take x equals negative one right here, as I approach from the right, this is gonna approach zero. So we'd say something like the limit as x approaches negative one from the right of g of x, this is gonna equal zero. If we were to approach it from the left, like so, if you approach it from the left, then on the graph, then your y-coordinates getting closer and closer and closer to zero as well. And so you get the limit as x approaches negative one from the left of g of x, this is what we would say is zero. Now you'll notice that the left-hand limit and the right-hand limit agree in this situation. And so that's what we mean by the two-handed limit. The two-handed limit exists if and only if the left-hand limit and the right-hand limit exist and agree with each other. So we say the limit as x approaches negative one of g of x, this is equal to zero. Now the curious thing is about this one right here, okay? This is an example of a removed point. We actually moved the point away and moved it somewhere else over here. You look at this, there's like this black hole, the point one comma two, seems like it ought to be on the function, but for whatever reason, this point's been moved up to one comma three. Why three, right? Well, you know, sometimes your parents are like, I want you to become a lawyer when you grow up. It's like, no, I love mathematics. I won't listen to you. You know, sometimes we can always live up to our expectations. Sometimes we can find something better, but for some reason this point has been moved away from the one comma two. Now in terms of the limit, the limit is looking for the expectation. If I were to pick some margin of error, then this value is gonna predict this right here because we actually can't look. You know, let me kind of do this in a slightly different way, erasing my screen a little bit. What we're doing when we see these functions, we actually can't see, we can't see what happens at x equals one. Aha, this is so clever, you'll never be able to tell. I know, it's super clever here. But if I were to put this on the screen, it's like, hmm, I can see that x equals one should be around here. What do you expect the function to be doing? Well, the function expects to be like if you're getting closer to one, you didn't be getting close to here. If you come from the other side, you seem to be here. So we see that the left limit as you approach one of g of x, it looks like it's gonna be a two. You know, if I was a betting man, that's what I would say. And if you take the right limit as x approaches one from the right of g of x, well, again, I would say it's gonna be a two, right? That's what you would expect. The left limit, the right limit, they both exist. They both agree. So we would say that the limit as x approaches one of g of x here, we expect that to be two. That's the expectation. But when we actually reveal back, right? We go visit our friend in the hospital and they injured their face. We expect that person when they take off the bandage to be Harvey Dent, but no, when they pull off the bandage, it turns out they're two-faced now. It wasn't what we expected. We couldn't have predicted that by looking at what happens near x equals one because the behavior near x equals one was predicting it was gonna be y equals two. But it turned out as y equals three. Who would have guessed? And so this is what limits are all about. Limits are trying to calculate the expectation, the trend of a function. They don't actually measure what the function actually does.