 Oftentimes, it's worthwhile to ask what is the in-behavior of a function? That is, what is the behavior of the function when x grows arbitrarily large? That is, x is getting bigger and bigger and bigger and bigger. Or in the other direction, what if x is growing arbitrarily small? That is going, going, going, going to the far left of the graph there. So really, in this lecture, we're interested in the idea of limits as x approaches positive or negative infinity. In our previous video, we had asked what happens, you know, what are the limit calculations when the y-coordinate gets close to positive or negative infinity? That gave us the idea of a vertical asymptote. In this video, we want to talk about limits, and we'll talk about this in many of the next few videos in this lecture series as well. What happens when you take the limit as x approaches positive and negative infinity? Well, oftentimes, as x goes to the infinite, y is going to go with it, right? As x approaches positive or negative infinity, y might also approach positive and negative infinity. That's a very common behavior for many functions. We'll talk about some of those in this very video. But there will be some times where this limit turns out to be a finite number, in which case, as x approaches positive or negative infinity, y approaches a number, not infinity or negative infinity. In that situation, we refer to that as a horizontal asymptote, and that's something we'll talk more about in the next video of this series. As we ask what happens as x goes to the far left or the far right of the graph, this is often described as the in behavior of the graph. How does the function behave as it gets to the very end on the far left or the far right? One example we can talk about would be a monomial function. Imagine n is some positive integer, so like 1, 2, 3, you got it. If we ask ourselves what's the limit as x approaches infinity of x to the n, well, the in behavior in that situation is going to be infinity. So if we were to draw this picture for a moment, imagine we wanted to draw y equals x to the n, just a typical monomial function, well, the right-hand side of the graph is going to look something like the following. It's just going to get bigger and bigger, bigger curving upward, and the amount of curvature will depend on the power to itself as the power of n itself. What you see in front of you kind of looks like half of a parabola, so that's why I said two, but as n gets bigger, it's going to get flatter right here and steeper right here, never actually becoming a vertical line, but it gets steeper and steeper, steeper. And so what we see for these monomial functions, y equals x to the n, is if we take the limit as x approaches infinity, then x to the n will become infinity as well. And this makes sense that if you take a big number and then you take a positive exponent of that big number, the number will get even bigger. Well, assuming that exponent is something larger than one, which for positive integers, of course, is the case. So taking powers of big numbers makes them even bigger, and thus as x gets closer and closer to infinity, x to the n will likewise get closer and closer to infinity as well. But when it comes to the left-hand side, that has to do with the parity of the power, that is whether it's an even power or an odd power. If you have an even power, the graph will be an even function. It'll be symmetric with respect to the x-axis. You're going to get a picture that looks something like this. It'll be a perfect mirror image right here, y equals x to an even power, in which case then, because the right-hand side does it, the left-hand side will do the same, you take the limit as x approaches negative infinity of x to the n, this will likewise be positive infinity. But on the other hand, if the power was an odd power, like we take x cubed or x to the fifth, this graph won't be an even function. It'll be an odd function. It'll be symmetric with respect to the origin. So you actually see the curvature going down here. So this would be y equals x to an odd power. Both sides will include the yellow piece right here, of course. And we see that as we take the limit of x to the n, as x approaches negative infinity, you end up with negative infinity right here. So like we talked about in the previous video about vertical asymptotes, when it comes to the in-behavior function, you can remember these statements about x approaching infinity and negative infinity with regard to the limits. But also, if you remember the graph, then the graph will tell you all the information. As you go to the right, the function goes up. That means as x approaches infinity, y approaches negative infinity. If you go to the left, the graph goes up in the even case. So as x approaches infinity, negative infinity, excuse me, y approaches infinity. Or as you go to the left and the graph goes down, this would say as x approaches negative infinity, y approaches negative infinity. That's all the limit statements are saying. We can remember this from the picture. So when it comes to monomials, the in-behavior is going to be simple like this. Now, of course, we can start modifying monomials. We can transform them, reflect them, stretch them, shift them around. We can add and subtract them with other monomial functions and make the family of so-called polynomial functions, the typical polynomial see right here. Now the good news about polynomial functions is that when you graph a polynomial function, and so let's do this real quick, if you were graphing a polynomial function, the in-behavior of a polynomial is going to look like the in-behavior of a monomial. What happens in the middle has a lot to do with those terms in play here. But the in-behavior of a polynomial function is just that of a monomial. And it's a specific monomial. The in-behavior of your polynomial function will just be the same as the in-behavior as its leading term. The leading term of a polynomial is going to be the power of x that's largest in the combination. That is, it's the largest power of x that doesn't have a zero coefficient. We'll denote this as a sub n times x to the n right here. n is the degree of the polynomial. a sub n, it's a number. It's called the leading coefficient. It turns out when you take the limit as x approaches infinity of a polynomial, this just becomes the limit as x approaches plus or minus infinity of a sub nx to the n. So the limit as x approaches positive or negative infinity for a polynomial will be the same as the limit of just the leading term as x approaches positive or negative infinity. I want you to interpret this as the in-behavior of a polynomial is determined by its leading term. And the idea in this situation, so let's take for example this example right here. Let's take the limit as x approaches infinity of x cubed minus 2x squared plus 1. Well, the largest power of x you see in that polynomial is x cubed. And so what we see from the previous statement is that the limit as x approaches infinity of x cubed, that's going to be the limit of this polynomial, right? We just determined it by the in-behavior. So we're approaching, as x approaches infinity, it turns out the parity of the power doesn't matter. It doesn't matter if it's even or odd. It just matters on the leading coefficient. If you have a positive coefficient, it's going to go off towards infinity. If you had a negative coefficient, it would go off towards negative infinity because you reflected across the x-axis. So this limit is going to turn out to be infinity. The polynomial x cubed minus 2x squared plus 1 is going to, its behavior on the right is it's pointing upward because we, because the leading term is. And why does the leading term matter so much? Well, the idea I want you to think of is the following, right? What, you know, what is a larger body of water, right? A bucket or a swimming pool, right? Well, clearly a swimming pool is much, much bigger, right? And if you were trying to like drain a swimming pool, just your typical swimming pool, say an Olympic-sized swimming pool, just so we have some type of clarity there. It's a pretty big swimming pool, right? As you take just like a standard five gallon bucket, maybe you get from like a home supply store, whatever. If you start, you know, taking that five gallon bucket and scooping the water out of the tank, right? Even though you're subtracting water from the tank, or from the swimming pool, it would take forever to drain that just by using your five gallon bucket possible. But the point is the magnitude of the bucket is essentially nothing compared to that Olympic-sized swimming pool. Well, now let's compare that to like say the ocean, right? The ocean is such a massive body of water that even if you were to take an Olympic-sized swimming pool, take a thousand swimming pools out of the ocean, you would see no difference whatsoever, right? People are very concerned about rising coastlines and such. Why is the solution not just, oh, have every citizen of California go take a bucket of water out of the ocean? Well, even that massive amount, like even if every citizen of California were to take a bucket of water, a bucket of seawater, that compared to the size of the ocean doesn't really make much of a difference. And so that's what we're trying to say right here, is that the growth rate of X cubed, when X starts getting big, big, big, big, right, is nothing, the growth rate of X cubed is so much larger than X squared that you don't even notice subtracting the two X squared from X cubed, because the difference is so much bigger. It's like taking a teaspoon out of the ocean. Sure, even if X squared is like, oh, that's growing faster, we're taking a swimming pool size amount of water out of the ocean. It still doesn't make much of a difference. The dominant term X cubed is going to be the fastest growing power function. And when X is really large, the other power functions in play, the other monomials in play for this polynomial are minuscule. They're dwarfed in comparison to the growth rate of X cubed. And so we continue this on. Let's look at the limit as X approaches infinity of negative two X to the fourth plus three X squared minus X plus two. Well, if you look at just the leading term, right, it's going to be negative two X to the fourth. Now, be aware, the leading term does not mean the first term. It means the biggest term present. So you need to make sure you look around just in case they're not in order. So if you take the limit as X approaches infinity here, the only term that dominates is going to be the negative two X to the fourth, for which then we want to ask ourselves what happens as X goes to infinity. Well, as X goes to infinity, well, by limit properties, you could pull out the negative two. And so we're going to get the limit as X approaches infinity here of X to the fourth, for which case then we see that, okay, as X goes to infinity, X to the fourth will also approach infinity. So you get negative two times infinity, so to speak. But what does it mean to multiply infinity by negative two? I mean, infinity is not a real number. Does it even make sense to multiply by anything? Now, I mean, the answer to this is going to be negative infinity. We'll talk about that more in just one second here. But the idea is if we didn't make this jump, right, if we didn't make that jump, we actually went directly to, whoops, the final answer right here. We could have said that because if you look at the graph of negative two X to the fourth, this looks like your standard X to the fourth graph. But the factor if you have a negative two, you've reflected it downward and then stretched it vertically, which in terms of infinity stretching, it doesn't make much of a difference, but the reflection does. And so as you approach the right-hand side, you're still going to negative infinity. So we don't need to multiply infinity by negative two, because again, the monomial negative two X to the fourth with its polynomial, its monomial growth and reflection tells us the right-hand side is going to go to negative infinity. But if we wanted this to be consistent with previous limit properties about we could factor out that coefficient, we end up with statements like negative two times infinity. How does one make sense out of such a thing? We'll get to that in just one more second, I promise. So another thing I do want to mention before we start delving into some dark magic in just a moment is what if we look at the reciprocal functions and ask what happens to them as X approaches positive negative infinity. We've talked about these functions already with regard to their vertical asymptotes, but these reciprocal functions also have horizontal asymptotes. That is as X approaches infinity or negative infinity, this graph is going to approach zero. So these graphs have a horizontal asymptote at the X axis. And so we can see here that if we take the limit as X approaches infinity of one over X to the end, this is always going to equal to D zero. And if we want to be slightly more precise, this will approach zero from above. You'll notice that the graph is above the X axis getting closer and closer to zero. Now if we take the limit as X approaches negative infinity of one over X to the end, this will be zero. That part's true, but I do want to break it up into even and odd cases because that seems to be the case we often see. If you take N to be odd, like we see in this picture, then you're going to approach zero, but you're going to approach zero from below. On the other hand, if you take N to be even, then you'll approach zero, but you'll actually approach zero from above. And again, this is an absence of any type of reflection of any kind. So if you want to see the even case, you're going to get something like this, like we saw previously. You'll approach zero from above. So one over X to the end, irrelevant to the power, will have a horizontal asymptote at the X axis. But how we approach it on the left-hand side will depend on the parity as well. And so basically what we can get away with is the following. We can make a statement like so. If I were just to plug in infinity here, we get something like one over infinity is equal to zero. If I were to divide by infinity, so to speak, I should get zero. Now it should be a little bit more cautious. If you take one over infinity, this should be looking like zero plus. And if you take one over negative infinity, that should look like zero minus. Like so, you'll notice in the even case, as X approaches negative infinity, you're taking an even power of a negative. So that's going to give you a positive. So the bottom is actually still looking like one over a positive number, which that positive number has large absolute value.