All right everyone, and let's get started with our limits at infinity, to infinity, and to calculus, I don't know. All right, here we go. So, let's start with an example. Let's take f(x)=1 over x squared. This is another one that should be on your list of graphs that you should know. Can you picture it? Can you picture it? No, maybe, I don't know. I remember this one is the volcano. Does something like this. So, obviously, you can't plug in at 0 because it's 1/0. That's bad. So, you get a nice asymptote to 0. And then it's x squared, so it's always positive. So you get this graph that looks like a little volcano to me. So, this is a graph, and I can ask questions like, what does the function want to do as x gets really large? What is the function? So, how do I write that, what's going on out here? And, as x gets really large, that's like saying x goes to infinity. What is the limit of this function as x goes to infinity? From the graph, you can see that this function approaches, but never quite gets to 0. So, I have a nice asymptote here. But that's okay. So, even though in the picture, in the graph, I never get to 0, the function wants to go to 0. This function will get as close to 0 as I want, although it never quite gets there. And that's perfect. That captures the idea of a limit. So, this limit, we'd say, is 0, even though you can't quote unquote, plug-in. Remember, infinity, this is important. Infinity is not a number. It's not like you point to the number line and say, aha, there's infinity. No, it's more of a concept. It's more an idea. This idea that numbers get larger, and larger, and larger. So I'm not evaluating this function at infinity. So, this is where precalculus and algebra doesn't know what to do with infinity. How do you study the infinite. But, limits are perfectly fine with saying, hey, and let x get large. Math and motion, let x be a bug, and it's moving on the graph, and it's going to the right, forever and ever and ever. Where does the graph want to go? Where does the function want to go? And it's important to get this down early and quickly, that infinity is not a number because then when you start seeing expressions or doing things, you can't just add it, you can't subtract it. Stuff gets funny with it. For all the same reasons, you can ask the question, what happens if x gets really small? What is the desires, the wishes of this function as x goes to negative infinity? So, where does that go? Well, down here as well. Same thing, asymptote, I also want to go to 0. In your head you can imagine, as x gets large, what effects were 100, or something like that? It's not that big, but you get the idea. So then this becomes 1/100 squared. That's a small number. What if x is, I don't know, 100,000? Well then the function becomes 1/100,000 squared. That's tiny, right? So this is slowly and slowly going to 0. Slowly, it's just going to 0. And so, you can imagine plugging in larger and larger numbers. As the denominator of a fraction gets large, the whole fraction gets really small. Over here, when I will talk about negative affinity because I'm squaring the function, the negative doesn't matter. So, it's not surprising these two things match, okay? So this is just an example that how you talk about infinity. I guess one thing to note, when you talk about limits, you can also talk about limits that go to infinity sort of up here on the Y-axis. So, what if I asked you this, the limit as x goes to 0? Let's go from the right, why not? 1 over x squared? Okay, so I'm a bug, [SOUND] there's my bug body, and bug head, and I'm walking on the graph to 0, from the right. This graph has an asymptote, I go up to infinity. I go off the volcano forever and ever and ever. So I would say this is infinity. You can put the +, it doesn't matter. Infinity + infinity. For all the same reasons, if I go to 0 from the left, now I'm a bug on this side of the graph. Walking my way up the graph, I also go to infinity. These two things match. So, what does that tell you? The overall limit, the two-sided limit, because they match, I can say this is infinity. Now remember, this is not a number. I'm not giving you back the number infinity. I'm just saying this graph, from both the right and the left, gets large. How large? Infinitely large, without bound. It goes off to infinity. It's a little bit semantics, but just, it's a special case. It's a special way for a limit to not exist. So, if I just said to you, hey, the limit doesn't exist, that tells you really no information. Maybe it's different from the right, different from the left. Maybe it oscillates between 1 and -1. But if I tell you, hey, the limit of this graph is infinity, I just get a little more information about the behavior of the graph. So this is a special case of DNE, again, not a number, just a specific case. Okay, now let's talk about vertical asymptotes. So vertical asymptotes are things that we've seen before, usually seem using calculators, but most students don't actually know what they are. They know when they see it, so, for example, we go back to our volcano [SOUND] can't talk volcano. There it is, our volcano, graphs something like this. There's a nice vertical asymptote at zero. There is this line at zero that something's happening. So if I ask students to tell me what's happening at zero, they'll say, there's an asymptote. I say, a great, what does that mean? They'll say, well, it's a line. What about the line? It's a dashed line, they get all excited. I say, well, that's not how you draw it, what's going on with the function? And they'll say, well, it's a line that you can never touch, and that's technically not true either. because I could just define this function as a piecewise function to be like 1 at 0. So they struggle, and the reason why students struggle is because they don't have the vocabularly to talk about asymptotes in terms of limits. So an asymptote, until you have limits, it's hard to talk about asymptotes other than just kind of point to them and say there it is. So we're going to define the line given by x equals a, to be or is a vertical asymptote, And spell it right to asymptote, good scrabble word. Vertical asymptote of y equals f of x, if, at least one of the following, Is true. All right, so if anyone of these three conditions that are to follow or true, I have vertical asymptote. And this is the real definition, not a line, dashline, or anything like that. So here we go, I need the limit as I approach the line or approaches equals a line to be either plus or minus infinity. So the function could go up or the function could go down, the point is the limit has to be infinity, when we're there. This is the overall limit. I'll say or, the limit as I approach a from the left, so this is the one sided limit of the function is plus or minus infinity. Or the limit as x approaches a from the right of the function is plus or minus infinity. This is the true definition of asymptotes and now you can see why this line x = 0 is an asymptote. As I approach a from right, left or both, it doesn't matter, this graph goes off the positive infinity, the limit is positive infinity. Therefore this graph has a vertical asymptote. We've seen lots of graphs that have asymptotes, let's just talk about a couple of the most common ones. How about f of x equals either the x, right, the most important function in calculus. If I draw this graph, start off low and I go high, there is an asymptote here at 0, so this is at y = 0. You'll have a horizontal asymptote. The most important function in calculus has an asymptote. Asymptotes are probably pretty important to study. Its inverse function because it's one to one, we saw this already. The logarithm also has an asymptote. And because the exponential has a horizontal asymptote, its inverse, which is the reflection of the graph, will have a vertical asymptote, at x is 0. Let's talk about this for a second. What is the limit as x approaches 0 of the natural log? What's the limit as x approaches 0 from the right? In this case, it doesn't really make sense to ask what is the overall limit, there is no 2 sided thing. If you did get asked this question, I would kick back and say that question doesn't make sense or this thing doesn't exist. So I want to just point that out that, because of the domain restriction you can't really ask for the two sided limit. However, it makes perfect sense to ask what is the limit of the natural logarithm as x approaches 0 from the right? Where does the function want to go? If you were a little bug, walking on this curve where you're headed off to as you walk from the right, that's often negative infinity. Because that's one of the conditions that's met, then we have a nice negative asymptote. And you don't have to be an exponential or a logarithm or anything else there's lots of other ones we're going to come across in this class. Just to show you that you can actually have more than one, we've seen this function before, but let's talk about it again. How many asymptotes does this guy have? So here is pi over 2, and here's negative pi, Ocver 2, so negative pi over 2 and this graph does this. But of course, this pattern repeats forever and ever and ever, right, does that thing. And there's another one back here. So there's actually infinitely many asymptotes here, vertical asymptotes in particular. And so you can talk about the limits as I approach. So it's the limit as x approaches pi over 2. They gotta be careful. Do you mean from the right or from the left? So let's do from the right. So now I'm a little bug and I'm headed that way of tangent of x and that's negative infinity, if I did the limit as x approaches pi over 2 from the left. So now, I'm a bug and I'm walking this way. Now I go off to positive Infinity and these are different. Positive, negative different. So the overall limit would not exist in this particular case. So you'll see a lot of these things be able to describe them and talk about there. Now just because if you notice, it's not immediately obvious if I have pi over 2 from the right or left that I'm going to involve infinity and that'll happen to sometimes. Infinity will sort of sneak up on you. He will not happens. Let's look at the limit as x approaches one from the right of the logarithm of the logarithm of x. Now I would imagine, most of us don't know the graph of this function, but that's oK. You could certainly go often graph this thing and then use the graph to figure it out, but let's see if we can be clever about it. We don't always want to run off to the calculator and try to figure this out. So the limit is x goes to 1 from the right. So remember, your order of operations here. Let's play around with the logarithm for a second. A logarithm graph come off on the side and be like what is the logarithm graph do? What's going on with this? So there's 1 and you gotta remember that the log of 1 is 0. Hopefully, you knew that. If not, write that down somewhere. The natural log of 1 is 0. So as x approaches 1 from the right, so coming in this way, where does this inside function want to go? I think it's like function wants to go to 0. So that's means that we could rewrite this question. So if I think of this is like going to 0, what's the way to to denote that? Think this is like going to 0. So if I come back and I write this thing using notation, I'm going to say x will now approach 0 from the right of the logarithm. The key here is what I don't want to see and this is we're going to get in trouble. If you write this, you're going to like a pointer notation. I'll put in red so you don't make this mistake. You can't say ln of zero. You can't just plug in like a lot of students will come off and be like this is just ln of 0. Remember, natural log of 0 is undefined. It's not in the domain. You just tried to evaluate a function at a point that's not in the domain. If you plug into your calculator ln of 0, I think your calculator blows up or something. I don't know. It just doesn't like that. And so you can write that. That's basically the same as writing like division by 0. So this is why limits are super important. So you got to write the limit every time and really mean like I'm just trying to understand the behavior of the function where does the function want to go. And now this second equation with one logarithm instead of two is a little more manageable as I approach zero from the right and a little bug, and I'm heading down here. This one I could just look at the graph. I know the graph of this one, this goes to negative infinity. So the final answer here is negative infinity and you hopefully can see y. And now go off and graph this thing and see what's going on. So use this thing to be a guide. So you'll see some examples that looks scary at first, but then just play around with them and try to go inside out to see where the function wants to go. Let's talk a little more about what it means for a function to have a limit at infinity and everything we say here will also be applied to negative infinity. So I'm asking about the behavior of a function as x gets large. Let's draw a picture and let's just not have it be 0 just because a lot of people sort of imagine a function that exponential is always a good one, something like that where it never touches. So here's a question for you. Is a function allowed to touch its asthma tote? Is it allowed to touch assessment? Can it be possible for the single touch? Most students when I ask that question they say no, they like no. It's forbidden there button, but that's not true. That's not true at all. For example, just to give you an example of a function. What if I had a function that sorted this do? Do, do, do. Do any got smaller and smaller and smaller and smaller? It oscillated forever, right? So here, we say the limit as x goes to infinity of the function is this number l. And over here Here, the limit as x goes to Infinity of this function is 0. Now, did this function touch is asymptote right? So here we have a nice horizontal asymptote at y = L. Here we have a horizontal asymptote, at y = 0. Does this function touch the asymptote? Yeah, infinitely many times in fact, so are you allowed to touch your asymptote? Absolutely, so you can touch it once you could touch it infinitely times by oscillation. All these things are horizontal asymptotes, but they're just different ways that a function can do it. You don't have to have infinite oscillations just to give you a picture of something that doesn't, maybe something like this, why not? And then it flat lines here. When you talk about infinity, remember, we only care what happens at the far right of the graph, or from negativity infinity the far left. I only care what happens here, and beyond. Whatever the function is doing before infinity, again, there are lots of numbers before infinity, it can do whatever it wants. It could be continuous, discontinuous, piece-wise touch once, touch none, I don't care. So we only care about what happens way out at the very, very large numbers and past all that. So just the overall trend, the overall behavior of this graph. So here you can see it touches at once,and twice and over here you get infinitely many times on this example, okay? So there's just some examples of limits infinity, so you can see limits are the tool to study infinity. That would be handy as we walk through this course. And remember the big takeaway here, infinity is not a number, so it's not going to behave like a number. It is a concept, it's a way of life, it's a philosophy. It's an idea that X is going to get really large. And we'll exploit that and play around with that, and see some of the cool things that calculus let's you do where algebra does not. Okay, great job, see you next time.
