 So what we're going to do to sort of get a visual of what the two different infinities look like is graph the function f of x is equal to 1 over x and we sort of did get our hands dirty and you know graphing some of these functions in series 3, right? But we're going to take a look at this and use a table of values to get a nice visual of what it means when the universe explodes, the language of mathematics collapses, and what it means to go on forever, right? To count until the end of time. So what we have to do whenever we graph in the function is write down our restrictions where we can't go, the impasses, right? And the impasse that we have in mathematics for the language of mathematics, the language that we use to understand the world around us is we can't divide by zero because we can't comprehend what happens. We don't understand what goes on if we ever end up dividing by zero and that means x would have to equal zero here, right? And x equaling zero would be our restriction and it becomes in mathematics when we're graphing something, it becomes our asymptote. It becomes an impasse and x equals zero is basically our y-axis here. So basically this line here is an impasse for us. We can't have x equaling zero. So what we're going to do is create a table of values for this function. So when x is equal to zero, f of x is undefined, right? So f of zero, I just put y there because this is shorter to write, right? So f of zero is undefined and that's our asymptote. Now what we're going to do is take a look at what happens in in the positive side of the graph, right? So what we're going to do is see what f of x is when x is 1, what f of x is when x is 2, when x is 3, when x is 4 and and then from there we're going to decide what the rest of the numbers are. So when x is 1, right? That's 1 over 1. f of 1 is 1, right? So y becomes 1. This point, if we're going to graph it, is going to be, let's say that's our 1, 2, 3, 4. So if that's one unit away, 1 and 1 is here, right? If we're graphing that point. When x is 2, f of 2 or y becomes a half. When x is 2, y becomes a half. When x is 3, y becomes a third, okay? And what's going to happen if you continuously increase x, it's going to happen, what's going to happen is exactly what we did in that in the first video we talked about, right? We're slowly going to approach zero. So if you continue to increase x, this line is going to approach y is going to be equal to zero. Okay, so if we continue to increase the x, this line here is slowly going to approach y is equal to zero, which is basically what we talked about in that first video of zero infinity. And this line here all of a sudden becomes an asymptote for us as well. The y equals zero line becomes a horizontal asymptote. Now we know how the function is going to behave from here this way, right? We have to figure out what the function is, how the function is going to behave from zero to one. And the way that's going to happen is the asymptote keeps on pushing. If a graph is moving towards an asymptote, the asymptote acts as a boundary, right? You can't touch an asymptote and you can't cross it. So the asymptote acts like a magnet. It keeps on trying to push this thing away. And what's going to happen is this line is going to get closer and closer and closer to the asymptote, but it's never going to touch it. So this line is going to go on forever, right? And the way that happens is, is if we start dividing one over x by numbers between zero and one. So if we divide, if we put set x is equal to a half, that becomes one divided by half. One divided by half is two. So when x is a half, y is two. So this line comes like this. And then if we take a quarter, one over one divided by a quarter, when x is a quarter, it becomes four and then eight and just continuously grows, right? And this is sort of a visual of what we talked about, the two different types of infinities. These lines here, the graph, right, when it goes on forever, you can visualize that. That just keeps on getting closer and closer and closer to the x-axis. And this one goes on forever and keeps on getting closer and closer to the y-axis. But the asymptotes themselves, we don't know what happens. We don't know what the function does or what happens to this thing at these points, when x is equal to zero and when y is equal to zero. And if we continue graphing this thing, what we're going to find out is, if we go to the negative x values, what we're going to find out is, we're going to get a mirror of that thing. It's just going to be down here. Okay? Because one, if we switch this with negatives, so if we switch these numbers and make all the x's negative, right, all that's going to happen is the y's are going to become negative, right? Because one over negative, one is negative one. One over negative two is negative a half, right? So all the y's become negative. So if we start graphing this thing, what it's going to look like is going to look like this. And again, the visual of going on forever this way, going on forever this way, and the impasses, the vertical and horizontal x and y asymptotes, right? Those are impasses. We don't know what this function does at those two points. And this is the way you can think about basically infinity, the two different types of infinities. One is the graphable type. One is the one where we can count it, we can visualize it, we can get values for it. And the other one, the asymptotes, the unknowns, the impasses are where the language of mathematics collapses. It doesn't give us any answers, right? There is no, we don't know what the question is. We don't, we have no idea what occurs, okay? And this is the two different ways that I look at infinity. And we'll continue talking about infinity and give different examples of how you can visualize infinity and start talking about zero in future videos. What we're going to do right now is take a look at what infinity, you know, try to visualize infinity in a different way, right? And this is something that I've used with a lot of my students. And for me as well, it was a, it was a wow moment for me when I looked at this thing, when I can't remember who initially introduced this, this to me, someone in some teacher that I had many, many moons ago, right? Or something that I read in the book, I have no idea. But it was, it was one of those things that made me go, wow, wait a second, what? You know, it blew me away. And that's what infinity is, right? It should, it should really blow you away. So for this, just to visualize infinity, just imagine us traveling from point A to B, but only going halfway every time.