 Funny. Let's graph the following function. I provide x is equal to let's go x over x plus 2. Do you want to x over x plus 1? No, let's go x over 1. Let's make it simple. We just want to look at the limits, right? So we're going to go, I wrote down x again. Let's go 1 over x plus 2, right? Uncharted days, how are you doing? Use Wolfram. Yeah, use Wolfram too. Hey Chico and Chad, hope you had a happy holidays. You too, you too as well, and happy New Year Uncharted days. So let's say we want to graph this, right? Graph this function. So the steps are the same, right? So first of all, you factor. If you're looking at a function, factor. Well, we don't need to factor this. It's already in simplest form, factored form. Then what you do is you find your restrictions. Both vertically, well, let's say you find your restrictions. We'll talk about asymptotes and stuff. So our restriction in mathematics is no dividing by zero. Really, that's the only restriction we have in mathematics. Uncharted days, if we have any more, let me know. But as far as high school mathematics is concerned, and they say restriction of no taking even roots of negative numbers, but that's not a restriction, that's dumbing down society, right? Induct like slowing down education, right? They used to teach it to us when I was in school, complex numbers, but they don't anymore right now in my part of world, right? So the only restriction we have in mathematics is no dividing by zero. So what you do, whenever you have rational functions or any type of function, you look at the denominator and if you can't divide by zero, then you say the denominator cannot equal zero, right? So this part, x plus two cannot equal zero, so x cannot equal negative two. That is your restriction for this function. So now that we're starting to get values, numbers, let's generate our graph. So here's our graph, right? Cartesian coordinate system, here's our x, here's f of x, which is really your y, and x cannot equal negative two. So here's, here's negative two, and what does it mean? x cannot equal negative two. Well, what you do with that is you say what's your asymptotes, right? So for the next thing you do after you find your restrictions is you find your vertical asymptotes. Your vertical asymptote, if x cannot equal negative two, your vertical asymptote is x cannot equal, x equals negative two, so vertical asymptote, right? It's just terminology, right? And gang, don't forget, freeassange, freeassange, freeassange. Julien Assange is a publisher and journalist that has been crucified for trying to bring transparency and accountability of capitalist power to humanity. For more information, see our Julien Assange and Wikileaks playlist. Marco Ciccio specifically, can you inform me on how to use limits to solve for horizontal asymptote? I'm going to show you right from here, right? You're going to see what limits means, right? So x equals two is a vertical asymptote. So this is, an asymptote is basically a boundary that your function cannot touch or cross, right? So your function can't equal x is equal to negative two. That means it can't, this is a no-go zone, this is a line, right? The universe explodes if you try to touch this line, okay? Now, that's your vertical asymptote. Your horizontal asymptote has these three things you have to consider, right? If you have a rational function, f of x, if you have ax to the power of n, the highest power, the degree on this thing, over bx to the power of n, the highest power in the denominator, okay? What you do is you say if n is greater than n, if the power up top is greater than the power in the bottom, if the degree, the highest power on the x is greater on top than it is in the bottom, then there is no horizontal asymptote. No horizontal asymptote. If n is equal to m, if the power up top is the same as the power in the bottom, then the horizontal asymptote is the degree in front divided by this degree, right? So the horizontal asymptote is going to be y is equal to a over b. And if the degree up top is smaller than the degree in the bottom, focus, focus, focus, focus. If n is less than m, then the horizontal asymptote is y is equal to zero, which is the x line, right? Which is the case that we have right now. Gang, thank you for the follows, by the way, and subs and donations and bits and stuff. If I'm missing them, I apologize. I just want to make sure I don't make any mistakes, no brain force while I look here and look here, right? So what we have here here is a horizontal asymptote. That means y, this is y equals to zero, the line y equals to zero. That means f of x can never be zero, which should be intuitive, right? Can you plug anything in for x to make f of x equal to zero? No, you can't. If you set x is equal to zero, that's one over two. Well, when x is zero, y is one over two, right? I really need to get better with mathematics. This just goes over my, does it on charger days? It's, once we start putting points on here, you'll see how this plays out, right? Now take a look at this thing. Now we need to graph this thing, right? We got our vertical asymptote, that's a no-go zone. We got our horizontal asymptote, that's a no-go zone, right? So what we have here, okay, is the boundaries of this function. And that's the way you start graphing a function. You find the areas that you can't go to, right? No-go zone, no-go zone, no-go zone, no-go zone. We got two of them, right? Then what you do is you try to find some key points, okay? First of all, we could do a table of values if you want. X and f of x, right? Let's do a couple of table of values. Now, we don't have a no-go zone when x is equal to zero. So let's find out what the y-intercept is when x is equal to zero, because that's negative one, that's zero, that's one, that's two, and etc., right? So let's find out what f of zero is. When x is zero, f of zero is going to be one over zero plus two, which is going to be one over two. Cool. So when x is zero, y is one over two. So if that's one, we're here, right? That point is on this graph. Now what you do is you ask yourself what happens to the function as you approach this asymptote, right? Now, the limits what you can do is we'll do it by point, right? But that's where calculus comes in, because calculus sort of gives you the behavior of a function that defines it better, right? So let's assume we're here, and we're going to start moving towards this asymptote. So let's find out what f of negative one is, right? So what we're going to do, we're going to go f of negative one is equal to one over negative one plus two. Negative one plus two is one, so this just becomes one. So when x is negative one, y is one, right? When x is negative one, y is one. Cool. Now, one of the properties of a function is when you're graphing something as asymptotes sort of act like magnets. If this is my function coming towards an asymptote, the function either does this because this sort of acts like a magnet pushing it, pushing it, pushing it, pushing it, right? The function comes like this, it either goes like this, right? Because it can't touch it, right? Or it comes in and goes like this, dives down, or comes in and does a, right? There's a couple of other things it could do too, right? So without calculus, you can only precisely estimate what a value is at a certain point. No, we're actually finding the exact value right now, right? We're actually finding what y is exactly when x is equal to negative one, right? And then what we do is we can get closer and closer to negative two. We can just put points here, points here, points here. But what's going to happen is basically this thing is going to shoot up, right? So what you can do is you can write it like this. You can say f of x as x approaches negative two from the positive side, right? Let me write this bigger so you see it, right? It's sort of terminology. Again, mathematicians are the laziest people on the planet. They create little symbols to figure things out, right? So if you've got a function, this function right here, as x approaches negative two from the positive side. So what we're saying is if we're on this function, we're going to approach negative two from above negative two. What is f of x equal to? Well, y goes up, up, up, up, up, up, up, up, up, up, up, up. It equals infinity. It goes forever. So it's going to be infinity, and you can say positive infinity if you want, because sometimes it might go down and become negative infinity, right? Now check this out. What happens as you go this way? As x gets bigger and bigger, so let's say x is two. If you sub in x is two, f of two is equal to one over two plus two, which is one over four. So when x is two, you're at a quarter. And because an asymptote, again, even a horizontal asymptote acts like a magnet here, let me put this down. So if this is a horizontal asymptote and we're coming down to this, this thing keeps on pushing up, pushing up, pushing up, pushing up, pushing up. It'll go closer and closer to this asymptote, but it'll never touch it, right? So what this means is this function goes like this. Why does it do that? Because no matter how big an x value you put in for x in this function, the bottom is going to become bigger and bigger, and one divided by a huge number is going to be really close to zero, but it's never going to be touching zero, and it will never be negative, right? So we're good on this side of the vertical asymptote. And what you want to do whenever you're graphing functions is you want to find out what's happening on either side of vertical asymptotes or holes or whatever you have, right? So you're mainly concerned when you're graphing something. The zones that you're going to look at are the x restrictions that you have, right? So vertical asymptotes that you have. So we know what the function is going to do on this side, and we know the function is not going to be here, otherwise it wouldn't be a function, right? Hello, Chico. How do you... How do you do? I do well. Thank you. I hope you're doing well as well. Happy New Year for Jamaica. Right on, Jamaica. Happy New Year, brother. Happy New Year, brother. Infinitesimal. It becomes infinitesimal. I hope you guys are having a nice warm Christmas here. You can hear the rain. It's gotten wintery, right? So we know what's going on on this side. Let's see what happens on this side, right? And by the way, the other way you could write for this is f of x. What happens to f of x as x approaches zero, right? Not, sorry, not zero. As x goes to infinity, right? As x goes to infinity, as x gets bigger and bigger, right? f of x approaches zero. Now, what happens as you approach negative two from the negative side, from this side, right? Is it going to go like this or is it going to go like this? Let's check it out. So let's find out, let's pick another integer closest, the closest integer to negative two, okay? No mods in the chat. Elder God is here. Yes, but how do we prove that using limits? How do you prove it using limits? This is sort of the proof. I don't know if it's, you call it a proof. What kind of proof? Oh, you're thinking about doing calculus in terms of, oh, okay, I'll show you, okay? I'll show you after we look at what the function does this way. Okay, you're talking about calculus, doing the first fundamental theorem of calculus, so we'll do it, okay?