 If you've seen the previous videos in this lecture series, then you've seen the contrast between the precise definition of the limit using epsilons and deltas. You've seen in this lecture number nine, more of an intuitive notion of limit as we see like points sequencing towards this limiting value. The intuitive definition I would argue is much easier and that's the one we're going to do the most often. It's much easier to get grasped on the calculation there. Why would we have the precise definition in the first place? If it's so much crazy harder and confusing, then this other approach would seem a lot easier. Well, it turns out this intuitive approach to limits does have some limitations. The prime example actually comes from the following. Consider the function y equals sine of pi over x. This is often called the topologist sine wave. It is based upon sine, but it behaves in a very different way because you have this division by 0 going on right here. You're dividing by x. I want to show you what happens when you get closer and closer to 0. Maybe we're interested in what's the limit as x approaches 0 of sine of pi over x. It's tempting just to plug in 0, but pi over 0 is undefined and so sine evaluated at a number that's undefined is clearly going to be undefined. We can't evaluate it. x equals 0 is outside the domain of the function sine of pi over x. But as we've seen in previous examples, just because the function's undefined doesn't mean the limit doesn't exist, there could still be an expectation of what the function should be doing. Let's consider that for a moment. If we were just to take some test points, let's just take a sequence of numbers that converges towards 0. Let's start with something simple. Let's do like reciprocals. If I take the sequence 1, 1.5, 1.3, 1.4, 1.5, 1.6, 1.7, you get the idea here. I keep on going. If you take these numbers and make the denominator just get one bigger each and every time, at the end of this long journey, you're going to end up with 0. Another way of thinking about that is the function y equals 1 over x. This function converges towards 0 as x approaches infinity. There's a horizontal acetote on this function. As x gets bigger, 1 over x gets smaller. We can approximate 0 by using this sequence of numbers. Seems nice and dandy in which case if we do that, the sine of pi over x will be defined for all of these numbers, like f of 1, that's going to be sine of pi which is 0. Sine of pi was a much interesting book compared to the sine of the beaver. Surprise ending right there, sine of the pi there. If we take f of 1.5, well, that's going to be sine of 2 pi which is 0, f of 1.3, that's sine of 3 pi which is 0, f of 1.4, that's sine of 4 pi over 0, excuse me, sine of 4 pi which is 0. Let's expedite this process. If I take f of a 10th, that's going to be sine of 10 pi which is 0. If I take f of a 100th, that's sine of 100 pi which is 0. If I take f of 1, 1,000th, that's 0. Not, sorry, not sine of 1, 1,000th, f of 1,000th here. I shouldn't be saying that. Sine of pi over 1,000. That's going to give us what we're giving right here. So if I evaluate the function, I want 1,000th to get 0. If we evaluate 1,000,000th, we're going to get 0. So if we just looked at this numerical data right here, and I can keep on going right, I could take numbers that are smaller and smaller and smaller and smaller. So if you give me any epsilon you want, epsilon greater than 0, then I can find a number, a small number such that, such that I'll give you a number n, we'll call it x, right? We'll give you some number x that x is smaller than epsilon, and then sine of pi over x is equal to 0. I can do that for some specific x value right here. So it's tempting, it's tempting that this numerical approach suggests that the limit is 0. Now, if you've been staring at this little picture on the screen while I've been talking to you, then this is actually the graph of our function y equals sine of pi over x. Now the software I use to compute, to generate the graphics for these lectures, you can actually see that it's kind of hit its limit when it comes to this graph. This is not the most accurate graph, but it gives you some idea of what's going on here, right? Because of the periodic nature of sine that as x gets close to 0, pi over x is going to get bigger and bigger, bigger, bigger, right? So like we're seeing right here, you can take sine of 100 pi, a million pi, right? And so what happens is because this thing oscillates, sine oscillates as normally as the angle gets bigger, bigger, bigger, it's oscillates every two pi. So as x gets smaller, smaller, smaller, the ratio gets bigger, bigger, bigger, and the rate at which it increases is actually growing rapidly. So you start to see that as you get closer to 0, the oscillation of this thing starts to go on steroids or something like that. I'm gonna switch this over to Desmos to give you a better depiction of what the topologist sine wave looks like. So when you look at this function right here, it seems all hunky-dory, right? When you get far away from the y-axis, it kind of mellows out as x gets big, because as x gets big, pi over x gets close to 0 and sine of 0 is in fact 0. So the asymptotic behavior of this thing will look like, it has a horizontal asymptote of the y-axis. What happens when you get closer to x equals 0, right? As I zoom in, you can start to see the bands are kind of pulling out. It looks like a really squished slinky right now. But as I start to zoom in, you can see that it looks like a blur, right? When you're right next to the y-axis, it looks like a blur, just a blob, right? There's no distinction between them. But on the other hand, when we start zooming in, if you take any spot of the blur, if you stare at it long enough, you can start to see the gap that's between the blur. It's not really a blur, it's just as the computer draws thickness to the curve so that you can see it. The thing is the curve is so close to each other that the thickness of the curve overlaps the gap between it. So it looks like they're overlapping, which is incorrect. And so as you continue to zoom in, in and in and in and in, closer, closer, closer, closer, you can always find a gap. But then when you look at this thing, it just looks like a blur, zoom in, zoom in. You can see these fine details, but again, it just looks like a blur over and over and over again. It doesn't matter how close you get, always will look like a blur. But when you zoom in more and more and more, you can find the gaps, right? The gaps are there between these functions, but the oscillation has gone crazy, okay? This is our topologist sine wave. And so when you look at this, when you look at the sine wave, for example, let's look at just the maximum elements for a moment, right? If you pick the right values of X, this place looks like one. This place looks like one, one, one, one. If we keep on going, keep on going, we have infinitely many points on the graph that look like Y should equal one, right? If we come back to the bottom of the graph, look here, there's infinitely many points. That's an infinity, right? There's infinitely many points on the graph that look like Y should equal negative one. Well, which one is it? Should the limit be one? But wait, I started this whole conversation thinking that maybe there's some sequence that goes towards zero, but heck, you can pick any number between negative one and one you want. And in which case, consider this, for example, take this sequence of numbers. All of these numbers, the Y coordinate is one half. So a negative one half, excuse me. And so therefore the limit should be negative one half. No, this function is much more interesting because there is no candidate for what the limit should be. The limit doesn't exist, but even worse, the left-handed limit doesn't exist and the right-handed limit doesn't exist. We don't have any evidence whatsoever. It's like, I shouldn't say that. It's not that we don't have any evidence. We have like a trial jury, right, going on and we have infinitely many witnesses and we have infinitely many different testimonies. How are we supposed to resolve that? There is no limit as X approaches zero. Now this situation is much, much more complicated, much more scary than the type of things we're gonna usually approach in this class. But it's because of things like this that we do need a precise definition of the limit. We can be very precise that this limit doesn't exist. And with the appropriate modification, what one could do is that this limit doesn't exist, right, this doesn't exist. But on the other hand, if I were to take the function, the limit as X approaches zero of X squared times sine of pi over X, this limit does exist. It's actually gonna equal zero. That's a topic we'll talk about another day. But because of sort of anomalies with functions because like this, we do need a precise epsilon delta definition of a limit to make sure we don't run into some type of bias we might have about a function. But the good news is for calculus one, these sort of biased situations as I'm describing right here are far in between. They're very rare. We actually will avoid them almost entirely. Therefore we can be successful for now with our intuitive notion of a limit. But ultimately when we go far enough it will have its limitations and eventually we will have to shed our hard scales then and take the more soft, the more comfortable position of this precise definition that is inside of this whole conversation.