 In our previous video, we introduced the idea of dominance when it comes to functions, right? How with power functions, the biggest power, the leading term is always the dominant term. With exponentials in logarithms, it gets a little bit more subtle, right? But we only dealt with functions that have sums and differences, for which you just added together a bunch of power functions, exponentials and logs and things like that. What happens when you start having like multiplication or in this video, I want to focus on ratios? When considering a ratio's in behavior, the in behavior will be the ratio of the dominant term in the numerator and the dominant term in the denominator. And this is something we saw when we had rational functions previously. So if I had something like y equals, we'll say like 2x cubed minus 7x plus 5 over, say like, we'll take 3x to the fifth plus 2x squared plus 3 or something like that. In this situation, we looked at the terms on top and bottom. So we have a 2x cubed on top, a 3x to the fifth on the bottom. So we'd say this function's approximately the same thing as 2x cubed over 3x to the fifth. If you simplify that fraction, you end up with 2 over 3x squared, for which as x goes to infinity, this thing will go towards zero, right, as x goes to infinity. And so this is what we called earlier, this bottom heavy function. As another example, let's say we have instead y equals, we'll take the same numerator, 2x cubed minus 7x plus 5, and this will sit above, we'll change the denominator. We'll take 3x cubed plus 2x squared plus 3. So I just lowered the power of the leading term there. For which then you see the dominant term on top is still 2x cubed. The dominant term on the bottom is a 3x cubed, for which that tells us that this will be approximately the same thing as 2x cubed. Over 3x cubed, which that fraction reduces down to be 2 thirds. And so that would then be the horizontal acetote. This thing approaches 2 thirds as x goes towards infinity or negative infinity. So, and this is exactly the case of what we called a balanced rational function. There was also the case of a top heavy, but that's because if you're top heavy, this ratio of dominant terms will simplify to be a monomial. And so it'll go off towards positive or negative infinity based upon the coefficients. So this is exactly what we were doing before. We just might have not realized it. For rational functions, there's just the three cases, bottom heavy, top heavy, and balanced. In more general ratios, we still do the same thing. We have to look for dominance. But we have to be careful though, because dominance can change its hand. You could pass the baton as you go from negative infinity to infinity. Dominance in general doesn't have to be the same as you approach infinity and negative infinity. For rational functions, whoever was dominant as you went towards infinity, will be the same as you approach negative infinity because of how power functions work. But again, it can get a little bit more subtle. So let's look at this example here. So f of x equals the cube root of 2x cube plus one over 5x to the fourth plus one. So let's determine the in behavior of this function. We'll do this by looking for the dominant terms. Ignoring sort of the cube root for a moment, just looking at the ratio, we have a 2x cube plus one over a 5x to the fourth plus one. Who's the dominant term? As x goes to infinity or negative infinity. Well, we see here that as x approaches, as x approaches, we'll start with that one here, as x approaches infinity, right? 2x cubed is the biggest power in the numerator. It's the dominant term. And then in the denominator, the dominant term is going to be this 5x to the fourth. And so what we can see here is that as x approaches infinity, f of x will approach, it'll be, it'll be asymptotic to, let's say that way, the cube root of 2x cubed over 5x to the fourth. Okay? And so that then would give, that's going to be the asymptotic term right there. And let me emphasize this is just asymptotic. So this is approximately equal to this. They'll be about the same thing. Now let's simplify that ratio. So you're going to get the cube root of, well, you have x cubed on top, x to the fourth on bottom. So this would simplify to be 2 over 5x, like so. For which if we rewrite it one more time, you're going to get the cube root of 2 over 5 times x to the negative one-third power, if you prefer. That's one way to do it, or you could keep with what we started with. Either one, I think would be perfectly fine here. We then have 4x cubed over 5x cubed over 5x cubed. We then have to ask ourselves, as x goes to infinity, what happens? So this friend right here, this 1 over x to the negative, or x to the negative one-third here, this is going to approach 0 as x goes towards infinity. Or you can see it right here, right? This is a bottom heavy, this is this bottom heavy ratio, right? The denominator has a degree bigger than the numerator. And so what's going to happen is as x approaches infinity, this thing is going to approach the cube root of 2 over 5 times 0. Excuse me, not 0 there, that would look like infinity. Which when you basically divide by infinity, we're not really dividing by infinity. We're just kind of getting some idea that as the denominator is bigger, bigger, bigger, bigger, the ratio gets smaller, smaller, smaller. This is going to look like the cube root of 0, which is equal to 0. So in summary here, what we've then mentioned here is as x approaches infinity, y will approach 0 right here. And we could also emphasize it's going to approach 0 from above, because this is going to be a positive quantity, right? The denominators can look like a positive infinity, the numerator is positive too. So this would look like we're approaching 0 from above, which can be helpful as we graph this thing. Great, that gives us, that gives us what happens as x approaches infinity. What would happen as x approaches negative infinity? Well, we have to be concerned about domains to some degree, right? So notice that as x approaches negative, or negative infinity, the denominator, the same basic thing is going to happen, right? Because there's no, there's no issue with domain here. Because you're taking cube roots, you can take a cube root of a negative, no big deal. If this is the square root, we would have some issues going on right here. We'd have to determine what the domain is. Because we wouldn't go all the way to negative infinity. But in this situation, we'll be C as as x approaches negative infinity, right? You might have a little bit more space here to write. As x approaches negative infinity, the same dominance is going to happen. As x approaches negative infinity, we still see that f of x will be approximately the same thing as the cube root of 2x cubed over 5x to the fourth, which that would simplify to be the cube root of 2 over 5x. And so now this will approach the cube root of 2 over 5 times negative infinity right here. So the reason I'm actually plugging in infinity here is so you pay attention to the signs, right? This time as we divide by negative infinity here, we're going to be getting 0 still, right? But this type of 0 is approaching 0 from below. And so this ends up giving us, we're going to approach 0. So the in behavior there is still 0. So as x approaches negative infinity, y will approach 0. But if we want to emphasize here, we're approaching 0 from below. So we have this horizontal asymptote. Let's describe it a little bit. We have this horizontal asymptote, which is the x-axis. Our function's going on the right-hand side, it'll approach the asymptote from above and from below, it'll approach the asymptote on the left-hand side. So we can get that, we can find that information just by looking at the dominant terms here. So let's look at a second example of this. We were looking for dominance and ratio. This one's a little bit different because we have these exponentials in play here. So if you ask the question as x approaches infinity, what happens? Well, the dominant term in the numerator would be 2e to the x. The dominant term in the denominator would be 3e to the x. And so what we see then is that g of x, as x approaches infinity, will be approximately the same thing as 2e to the x over 3e to the x, for which e to the x is common divisors cancels out, and we see that this right here is just going to be 2 thirds. And so this then gives us a horizontal asymptote. We have a horizontal asymptote at y equals 2 thirds, okay? That's if we approach the right-hand side. When we approach the left-hand side, though, we have to recognize that there's a difference as opposed to the power functions we saw on the other side where dominance doesn't change. For exponentials, dominance does change as you move from one side to the other. Because when you look at e to the x, for example, its graph would look something like the following, right? On the right-hand side, it's going to approach infinity. But on the left-hand side, it's going to approach zero. So as we approach, as x approaches infinity, e to the x became dominant in that situation. But as you approach negative infinity, e to the x, these things are going to disappear, right? This thing right here, these things are just going to go off towards zero. And in fact, the dominant terms, as x goes to negative infinity there, is going to be the constants. Because the constants don't go off towards zero in that situation. So as x approaches negative infinity, we see that g of x will be approximately one-seventh. Because again, the e to the x's will vanish as x goes towards negative infinity. And so this graph is going to have two distinct horizontal acitotes. On the right-hand side, if we were to sketch the picture here, on the right-hand side, we have a horizontal acetote, which is y equals two-thirds. But on the left-hand side, we have this horizontal acetote at one-seventh. That's a positive one-seventh, isn't it? So we get something like this. So we can anticipate that our picture is going to do something like the following maybe. On the right-hand side, it goes towards two-thirds. On the left-hand side, it goes towards one-seventh, like so. And so when you look at ratios, you want to look at the dominance as you go towards infinity and negative infinity. And you need to recognize that the approach towards negative infinity could be different than the approach towards infinity. Because the baton of dominance might shift into a different hand when you approach negative infinity. The thing is, as we talked about rational functions earlier in this lecture series, that's not something that happens with power functions. Power functions never pass the baton. But exponentials do. They're much more generous. They're much more team players in that regard. And so you have to watch out for logarithms and exponentials because the baton of dominance can shift as you consider positive infinity versus negative infinity.