Hi, everyone. 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 of X =1/X squared. This is another one that should be on your list of graphs that you should know. 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 zero, because it's 1/0. That's bad, so you get a nice aspen total zero 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? Like 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 zero. So, I have a nice asthma toad here, but that's okay. So even though in the picture in the graph, I never get to zero, the function wants to go to zero. This function will get as close to zero 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 zero. 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. Know 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 dysfunction at infinity. So this is where like pre calculus and algebra doesn't know what to do with infinity. How do you study the infinite, but limits are perfectly fine with saying, hey, let X get large math and motion, let X be X 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 I 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, like you can't just add it, you can't subtract it. It's stuff gets funny with it, for all the same reasons you can ask the question with this graph. Like 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 asymptotes. I also want to go to zero. in your head you can imagine, like as X gets large, what effects were a 100 or something like that. It's not that big, but you get the idea, so then it becomes one over 100 squared. That's a small number, what effects is like, 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 zero slowly, it's just going to zero. 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 infinity 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 ask you this? The limit as X goes to 0? Let's start with, let's go from the right, why not one over x squared. Okay, so I'm a bug due to do, there's my bug body and bug head and I'm walking on the graph to zero from the right. This graph has an asymptotes, I go up to infinity. I go up the volcano forever and ever and ever. So I would say this is infinity, you can put the plus, it doesn't matter infinity plus infinity. For all the same reasons, if I go to zero from the left now I'm a bug on this side of the graph walking my way up to 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. Remember, this is not a number, I'm not giving you back like the number infinity. I'm just saying this graph from both the right and the left gets large. How large, infinitely large without bounds. It goes off to infinity. It's a little bit of 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 like, hey, the limit doesn't exist, that tells you really know information. Maybe it's different from the right, different from the left. Maybe it oscillates between one and negative one. But if I tell you, hey, the limit of this graph of infinity, I just get a little more information about the behavior of the graph. So this is a special case of DNA again, not a number, just specific case. Okay, now let's talk about vertical asymptotes. So vertical asymptotes are things that we've seen before. Usually seen freak out, but most students don't actually know what they are. They know when they see it, so, for example, if we go back to our of our volcano can talk volcano. There it is, our volcano graph, something like this, there's a nice vertical asymptotes 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 great, what does that mean they'll say, well it's a line, what about the line? It's a dash 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 like you can never touch and that's technically not true either because I could just define this function as a piecewise function to be like1, 0. So they struggle and the reason why students struggle is because they don't have the vocabulary to talk about asymptotes in terms of limits. So, an asymptotes 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 as if vertical asymptotes. Yeah, spell it right, asymptotes good scrabble word, vertical asymptotes of Y equals F of X. If at least one of the following, it is true, all right. So if any one of these three conditions that are to follow our true, I have vertical asymptotes and this is the real definition not, a line dash line or anything like that. So here we go, I need the limit as I approach the line or approached X equals the line to be either plus or minus infinity. So the function could go up where 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 equals zero is an asymptotes as I approach a for right, left or both, it doesn't matter. This graph goes off to positive infinity, the limit is positive infinity. Therefore, this graph has a vertical asymptotes. 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 e to the X right? The most important function in calculus, if I draw this graph, start off low and I go hi, there's an asymptotes here at zero. So this is at Y equals 0. You have a horizontal asymptote. The most important function in calculus has an asymptotes. Asymptotes are probably pretty important to study its inverse function because It's 1 : 1, we saw this already. The logarithms also has asymptotes and because the exponential has a horizontal asymptotes, its inverse, which is the reflection of the graph, will have a vertical asymptote at X zero. Let's talk about this for a second. What is the limit, as X approaches zero 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 two-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 you have to 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 logarithms? As X approaches zero from the right, where does the function want to go? If you were a little bug, walking on this curve where you headed off to as you walk from the right, that's off to negative infinity. So because that's one of the conditions that's met, then we have a nice photo plasma toad. And you don't have to be an exponential or a log a rhythm 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's pi over two and here's negative pi over two. So negative pi over two 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, infinitely many asymptotes here, vertical asymptotes until it's in particular. And so you can talk about the limits as I approach. So it's the limit, as X approaches pi over two, you 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 two 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 infinity, 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 their now, just because if you notice it's not immediately obvious if I have pi over two from the right or left. That I'm going to involve infinity and that will happen to sometimes infinity will sort of sneak up on you. I hate when that happens. Let's look at the limit as X approaches one from the right of the logarithms of the logarithms of X. Now, I would imagine most of us don't know the graph of this function, but that's okay. You could certainly go off and graph this thing and then use the graph to figure it out. But let's see if we can be clever about it and we don't always want to run off to the calculator and try to figure this out. So the limit is exposed, the one from the right. So remember your order of operations here, let's play around with the log rhythm for a second log rhythm graph come off on the side and be like what does the logarithms graph do? What's going on with this? So there's one and you got to remember that the log of one is zero. Hopefully, you knew that if not write that down somewhere, Tthelog or natural log of one is 0. So as X approaches one from the right, so I'm coming in this way, where does this inside function want to go. And the inside function wants to go to 0. So that's means that we could rewrite this question as if I think of this as like going to zero. What's the way to denote that. Think of this as 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 zero from the right of the logarithms. The key here is what I don't want to see, and this is where you're going to get in trouble. If you write this you're going to get like a point for rotation, I'll put it in red. So you don't make this mistake. You can't say In of 0, you can't just plug in. Like a lot of students will come off and be like, this is just In of 0. Remember, natural log of 0 is undefined, it's not in the domain. You just evaluated, you just tried to evaluate a function at a point that's not in the domain. If you plug into your calculator, In of zero, I think your calculator blows up or something. I don't know, it just doesn't like that, and so you can't write that. It's basically the same as writing like division by zero. So this is what limits are super important. So you've 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 logarithms instead of two is a little more manageable as I approach zero from the right, I'm 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, negative infinity. So the final answer here is negative infinity and hopefully I can see why and now go off and grab this thing and see what's going on. So use this thing to be a guy. So you'll see some examples that look 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. Now a lot of people let's draw a picture and let's just not have it be zero. Just because a lot of people sort of imagine a function that, the 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 asymptote. Is it allowed to touch his asthma, can it would be possible for this thing to touch. Most students when I asked that question, they say no, they're like no, it's forbidden verboten, 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 sort of did this do do do do do. And it got smaller and smaller and smaller and smaller, it oscillated forever. All right, so here we say the limit as X goes to infinity of the function is this number L. And over here, the limit as X goes to infinity of this function is 0. Now, did this function touch, it's asymptotes, right? So here we have a nice horizontal asymptotes at Y equals L. Here we have a horizontal asymptote at y equals zero. Does this function touch the asymptote? Yeah, infinitely many times, in fact. So are you allowed to touch the asymptote? Absolutely, so you can touch it once you can touch it includes many times by oscillation. All these things are horizontal asthma toads. There's 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 negative infinity, the far left. I only care like 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, peaceful eyes 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, twice and over here, you've got 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 will 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 well, exploit that and play around with that and see some of the cool things that calculus lets you do where algebra does not. Okay, great job, see you next time.
