Alright, hi everyone, and let's get right into it, today we're going to talk about limits at infinity. Infinity remember is this idea, it's not a number and it does not behave like a number. Just to give you an example, if I start talking about, the limit as x goes to infinity of, let's take an easy function here, e to the x, e to the x this is a function we should know, it goes low to high. I'm really asking, across the watercourse, I'm really asking, what is the function doing? How is the function behaving as x gets large? Now when I phrase the question that way, how is it behaving? What does it want to do? Hopefully you don't come back and say seven. It doesn't make sense to think of infinity as well, when x is seven, this is what happens. No it's, x is moving, x is going along the X-axis, going off in the positive direction, what happens to the function? In this case over here, we'd say the function would go to infinity, it increases forever. You can also talk about x as it moves in the negative direction, where does the function want to go? We talked about this before, we have nice asymptotes, here the function wants to go to zero, It never gets to zero, but that's the difference between evaluating the function and actually talking about limits here at zero. There's lots of functions that go off to infinity or negative infinity or actually approach a number, one of the main types that we're going to talk about first are what's called rational functions. But it's super important if you see stuff, here I'm going to put in red, that means I'm serious when I put it in red, if you see things, for warning, rookie moves that are gonna come up in calculus, if you see things like infinity divided by infinity, like as you take a limit, we're going to use this notation, it's terrible, this is garbage. You can't cancel and be like, "Oh that's one," no because infinity is not a number. What if I have things like zero over infinity, is that zero? No, once you put infinity as a symbol, you're more talking about an idea, a concept or something. In particular, the one that's going to drive me crazy I see this all the time, if I talk about two, let's say I have two functions and they're both going to infinity, e to the x minus x squared, and they're both going to infinity. If I look at the behavior, well this one's going to infinity and this one's going to infinity and I subtract them, can I just say that this is zero? Like infinity, no, no, no, no, no. Think of it like two trains off to the positive direction, their difference, the trains may over time fall apart, they may come together, they may reach a finite distance, it's math in motion. You can't, when working with infinity, just treat it like a number and say, "Oh this is going to be zero," no, no, no, no, no, we'll see an example of that in a minute. Just be super careful when you have infinity and you're working with it, it's a dangerous place to go, don't go there unsupervised, be careful, proceed with caution. The first example of functions that have a nice behavior with infinity is our rational functions, for example, let's just start off with an example here, the limit as x goes to infinity of x cubed plus 5_x over 2_x cubed minus x squared plus 4, let's look at this for a second so rational function, the degree at the top is three, the degree of the bottom is three as well. I'm going to do this algebra, it's a little annoying, but we're going to do it once. Just take this, it's a little bit of a trick, I guess, but the leading coefficient, the highest degree that I see is x cubed, is x cubed. So what you can do, I'm going to come out of the limit and rewrite this, I don't want to break any of my notation after I yell at you guys to have good notation. Here's what I'm going to do, and this is one of those moments in calculus where I take something and I make it look worse before it actually looks better. I'm going to divide everything by a fancy number 1, 1 over x cubed. We've seen this before some times where you make stuff look worse before it gets better, it's not immediately obvious why I would do that. Hopefully you agree that I'm just multiplying everything here, like the top and the bottom, this is a fancy way to write the number one, so I am multiplying by one, I'm not actually changing the problem, but why would I write it this way? Well you'll see in a second, let's distribute the one over x cubed to both pieces, and you get x cubed over x cubed plus 5_x over x cubed, that's our numerator we do on the bottom, we get 2_x cubed over x cubed minus x squared over x cubed plus 4 over x cubed. All right, I've definitely made this thing look worse and look terrible, but you can start to see that some cancellation is occurring, for example, x cubed over x cubed becomes 1, this becomes 5 over x squared, that's my numerator, the denominator becomes 2 minus 1 over x plus 4 over x cubed. Okay, so instead of looking at the rational function in its original form, I look at it this way. There's a reason why I do this. There's a madness to the method. As I take the limit to this expression. Take the limit as x goes to infinity of one plus 5x squared, over two minus one over x plus 4x squared. We're going to use the limit laws. Remember limits are beautiful. They distribute over addition, and they distribute over subtraction, and they distribute over division. Assuming the bottom is not zero, which we'll see in a second it's not going to be. You can think of this as one plus the limit, as x goes to infinity of five over x squared. Instead of treating the entire function as a single thing, you look at each piece. One minus the limit, as x goes to infinity of one over x, and then plus the limit here. How does each piece contribute to the overall limit? Now, one over x squared, and so again the five, you don't care about it actually it moves out front. One over x squared and others, one over x as well we've seen before. Let's do a picture. One over x squared, remember that's the volcano. Looks like that. One over x is just the flip like this. These are the graphs of one over x squared over one over x that we should know. At infinity, as x gets large, they have something in common, they both go to zero. Again, the five comes out front, who cares? It's one over [inaudible] This term goes to zero, the five over x squared. The limit as x goes to infinity of one over x, that also goes to zero. The limit as x goes to infinity of four over x squared also goes to zero as well. In that case, what are you left with? One plus zero over two minus zero plus zero. A lot of zeros in there, clean that up, you get one-half. Using the limit laws, which is perfectly suited for rational functions, we have a nice limit of one-half. I want to show you something here. This is not going to be a coincidence. This will happen for all our rational functions. Because the degree of the top and the degree of the bottom are both three, they're both the same. This function, these are called the leading terms, this function is completely controlled in the limit by the leading terms. When that happens, the function behaves like x cubed over, two x cubed. As we saw with the limits, these other terms go to zero. They are not contributing to the overall value of this function at infinity. Since the overall function behaves based strictly on its leading terms, these x terms cancel. We're saying this function in the limit, behaves like the function one-half. You can think of it. Sometimes people say it's the ratio of the leading terms if there's an invisible one there. The limit as x goes to infinity, when the numerator and denominator have the same degree, is the ratio of the leading terms. There's a bunch of ways to see this. Let's do another example where the degree of the top and the degree of the bottom are not the same. Let's do an example where we're going off to infinity. Limit as x goes to infinity of, x cubed plus 5x over x squared plus four. For all the same reasons, I'm going to shorthand this one a little bit here, this function behaves [inaudible] like x cubed, over x squared. When you go into infinity, what's bigger, x or x cubed? x cubed. The 5x does not determine the behavior of the numerator. For the same reason, when you go to infinity, we pass four very quickly. Four does not determine the behavior of the denominator. This function in the limit, this is key to realize, this squiggle mark is not an equals sign. I'll write the limit here so you see it, it's equal to the limit as x goes to infinity. Because the two leading terms are x cubed and x squared, I can clean that up and say this is the limit as x goes to infinity of just x. y equals x. We know that graph, it's a line, goes through the origin [inaudible] off to infinity. This limit equals infinity. This function will grow larger as x gets larger. For all the same reasons, let's do another one. The limit as x goes to infinity. This was with degree on top was bigger. Let's do one now where the degree on the bottom is bigger. Take 5x over x squared plus four. x is going to infinity. I have a nice rational function. That four is -, it doesn't do anything. I'm off to infinity, four see you later. This function will behave like the limit as x goes to infinity of 5x over x squared. Again, if you want to see the 5x over x-squared, do what I did in last example, and divide every term by x squared, and bring in the limit and use limit laws. I'm just trying to get the big points here, so we don't get lost in the arithmetic. This function behaves in the limit like its leading terms. 5x upstairs numerator, x squared downstairs. Clean that up and cancel, and what happens? You get the limit as x goes to infinity of 5 over x. The 5 doesn't matter. Don't get hung up on the 5. It's a constant. I care really about what's happening with x, 1 over x, drawing this graph a lot. Hopefully, picking up this one quickly, as I go to infinity, where does the graph want to go? To 0. So this thing goes to 0, and I'm left with 5 times 0. This graph is 0. The limit of this 1 is 0. So these three examples are just prototype examples. The big picture, the big takeaway of this is, rational functions are completely determined by the relationship between the degrees, of the numerator, and the degree on the bottom. Let's summarize this as we go here. So take a rational function f of x equals, sum p of x, over q of x. Over q of x, a rational function. Remember, rational function means the top is a polynomial, and the bottom is a polynomial. So you have to be a polynomial for this to work. If the degrees are equal, let's start with that. If the degree of the numerator is equal to the degree of the denominator, then what happens? Then the limit, as x goes to infinity, of f of x is equal to the, I'll say ratio of leading terms. Ratio of leading, I should say coefficients. Ratio of leading coefficients. Whatever the coefficient is on the numerator, divided by the ratio of the leading term in the denominator, that's what's going to happen. If the degree of the bottom, let's say degree at the top, is bigger than the degree at the bottom, think of it like a tug of war. As I go to infinity, they're fighting for control. Higher degree wins. If the numerator is bigger, the nominator might as well not even be there. So if I'm off to infinity, then we'll say this limit, the limit here, as x goes to infinity of the function, will equal infinity. It'll go off to infinity. If the degree of the denominator is bigger, so we'll write it like this, of q of x, then the denominator wins. The denominator wins. So we'll say, that the limit of the function will go to zero. Watch out for small variations of this, where I have a limit go into negative infinity. That just means x is negative. So you got to play around with it. Like, if it's squared, it won't matter. If it's odd, you may get negative infinity for the second one here. So just watch out for that. So plus, or minus infinity, I should say, just to be a 100 percent sure. But it does not matter if the degree at the bottom, it will be bigger. One of the things you can think about is, just how does a fraction behave? As the denominator gets larger, the fraction gets smaller. One one-tenth, one one-hundredth, one one-thousandth, one one-millionth. As the numerator gets larger, the denominator gets larger as well. So they have this inverse relationship. These are how rational functions behave. Just to show you an example of some crazy stuff that can happen. This is an example, and I'm going to put a little star next to this one. It's tricky. So a little tricky, and maybe even counterintuitive, and that's the dangers of infinity. We have nine x squared plus x, minus 3x, 9x squared. So stare at this for a second. I imagine we don't know the graph of this, so we're going to attack this with purely algebraic means. This is where I yelled at you the last time. Just be careful, don't make this mistake. If I plug in infinity into square root of 9x plus 6 square, I mean, plug in, you know what I mean. If I think of a large number, what's be happening? The leading term here is 9x squared, that's like going off to infinity. If I add more, bigger, larger numbers, I'm getting super large numbers under a square root, and the square root graph also goes off to infinity, so this whole term here is it wants to go to infinity, this first term, minus 3x. Now the 3x itself is just the line I'll say minus, we just considered 3x, it's a line, and it also wants to go to infinity. You'll see this infinity minus infinity. The mistake to make why this is such a hard question is because it's counterintuitive, you probably used the numbers, and you want to say infinity minus infinity equals 0. Don't do that. I'm going to put it in red. It is a mistake, and the very common ones would just be like, "Oh, this is zero, I am done." Then you get no points. We've seen before how to handle, or how to work with square roots. If you remember on our list of techniques to work with limits, one of the techniques we have, for square roots in particular, is to take conjugate. I'm going to do that again, hopefully, you pick it up when to use the conjugate. Conjugate says put a plus 3x over here, and whatever you do to the top, you must do to the bottom 9x squared plus x plus 3x. I'm multiplying by a fancy number 1, so if you agree, I'm not changing the question. Now, we'll take the limit as x goes to infinity. Numerator, I think if this over 1 becomes foil. First is 9x squared plus x. Outside, inside, cancel, then, last becomes minus 9x squared, and that's nice, because these two terms, the 9x squared and the minus 9x squared will cancel. In the denominator, we just multiply the denominators together, there's not much going on here, 9x squared plus x, plus 3x. Is this a little better? It will turn out the answer is yes, and one of the reasons why, is because, the 9x squared cancel, so I have a nice x upstairs. If I factor out some stuff downstairs, what am I left with? Let me just clean this up. Limit as x goes to infinity of x, over the square root of 9. Let's write this x here, x squared, what's left? Messed that up. Try again. I'll just write it, 9x squared plus x, plus 3x. Plus 3x is downstairs but it is not inside the square root. Let's divide it by 1 over x. So what's this? Limit as x goes to infinity, we'll divide everything by x, 1 over x times. I'll just write it here, 1 over x, 1 over x, and the reason why we do that, is because now the numerator, becomes one, and I have 1 over x, but I want to write 1 over x, I want to play around to square root. I'm going to write it as, 1 over the square root of x squared. Hopefully, you agree that 1 over the square root of x squared, is in fact, 1 over x. The reason why I do that, is because I have to play nice with the square root, and you can multiply square roots together. I'm writing this in a weird way, but everything is legal so far. I have a one upstairs, and 1 over square root of x squared. Let's bring this in, to both pieces. The reason why I have the square roots, is because now I can work inside the square root, so the square roots combine. I'm using the fact that square root of ab, is square root of a, times the square root of b. If you have two square roots multiplied together, you can work underneath a single square root. Basically, I can divide 9x squared by x squared, and I can divide the x by x squared as well. When I distribute this 1 over x, and 3 over x, I get left with a plus 3. Clean this up, keep going, almost there. Hang in there. X goes to infinity, we get 1 over, told you it was tricky, so what do we have here? The square root of 9 plus 1 over x, plus 3. Limit laws apply. You can bring the limit into each piece, the only thing that has an x in it is this 1 over x, and we're going limit as to infinity. So I've drawn this graph a lot, one over a very large number that tends to zero. Finally, finally, finally, finally, what am I left with? Square root of 9 plus 3, that, of course, is 1 over 3 plus 3, better known as one-sixth. The final answer here, this is super weird. Infinity, minus infinity, turns out to be one-sixth. Now, of course, that's not true. It could be any number you want, but think of it as two trains, they're moving and over time, their distance, becomes one-sixth in the limit. It is not obvious at all, looking from this problem, that you're even going to get a number, let alone an infinity. There are plenty of differences where you can get infinity, this just happens to be one, where you get one-sixth. This is a tough example. A good application of the conjugate to solve a limit, so go over this one again, lots of good algebra here and practice that. Let's do one more set of examples. These are going to be a little more straightforward. The way to get better at these that play around the infinity, is to just do examples, and I would assume not too much, and make sure you have good reasoning as to what you're doing with these things. We'll do, let's say four of them here. Here we go. I'll give you some different cases. Limit as x goes to infinity of e_minus x square. If you know the graph of this one, e_minus a positive number, that's exponential decay. There's a shortcut around this one, if you know the graph. So e_minus x squared. It is at least, well, this part is what I care about. As I go to infinity, this thing is going to get small. The answer to this one is going to be zero, and you could argue by graph. If you want another way to do this, so you can say zero and drop, I mean, a little the graph here, talk about it. Another way to think about it also, e, the base function is the composition of e and negative x squared. e is a continuous function so if you want, you can move the limit inside. This is by the theorem with no name. You can move the limit x squared inside. That is allowed because e and negative x are continuous functions. Now you have an easier limit to work with, this is x squared, this is the parabola, this is the one everyone knows, as I go off to infinity, x squared goes to infinity. This becomes the limit, this is the long way to do it, as x goes to infinity of e to the minus x. You can replace the answer to this limit, cancel off every squared with an easier limit. In that case, e to the negative x, that is exponential decay over time that goes to zero, you use the graph to see this and you can write out zero. Both of these work, the intuition hopefully is e to the negative number over time goes to zero. Let's do another example. How about the limit as x goes to infinity of cosine of x? Here your knowledge of the graphs of these functions is critical. Cosine of x that oscillates, I'll just draw the right side, since that's all we care about. This thing oscillates forever and ever. If I asked you, where does the function wants to go, what is the number that the function wants to go to? As it gets large, there is no more number, it goes to one and negative one, one, negative one and bounces back and forth. This example here we'd say this does not exist, it oscillates if you want to sound fancy and smart, it oscillates between 1 and negative one. Then you can use the graph to show that. In general, unless it's a very obvious graph from precalculus, you can't use the graph, but you can still do that for this one. Let's do another one. Cosine is easy enough. Limit as x goes to infinity of e to the minus 2x cosine x, e to the minus x cosine x. This one's a little tricky. If you want pause the video on this one, if you want to take a break here to try this one, e to the minus 2x. If you remember, you can't just "plugin," you can't do algebra or foil factor. If you go down your list, you're running out of things to try, and you finally get to squeeze theorem. Remember squeeze theorem is out there and I see something with cosine, so that makes me want to bound this function. Cosine is bounded above by one and below, by negative one, so you can bound this function as minus 1 times e to the minus 2x, and here's the function in question, and then you can bound that above by e to the minus 2x. Cosine is bounded above by 1, below that I'll place 1 and negative one on the upper and lower bound. Now, the limit as e goes to infinity, these are exponential decay, it's like we had before. E to the minus 2x, that's exponential decay that goes to zero, and e to the minus 2x, this actually goes to negative zero technically, but of course that's just zero. I have some limit that is greater than zero and less than or equal to zero, so by the squeeze theorem, has a name, so excited, by squeeze theorem this is equal to zero. This limit is also equal to zero. One more example, I clear the slide here. For example, let's take the limit as x goes to negative infinity of x to the fourth plus x to the fifth. I don't love these questions, they're a little tricky, but you'll see it once in a while. Be careful just because they write it as x to the fourth first, that's not the leading term. What's higher 4 or 5? Hopefully you agree 5 is higher. The leading term is here. The leading term, even though it's not written first. If it bothers you, you can certainly rewrite this thing as x to the fifth plus x to the fourth. This is the function, it's going off to infinity. The leading term determines exactly where this thing is going to go, the x to the fourth, you might as well just pretend it's not even there. This is weird, you can't normally just erase stuff the algebra police will yell at you. But in the limit, all we care about is the leading term, that x to the fourth matter as we'll not even be there for all we care, and so x to the fifth now you have to know what odd functions look like. They go low to high, and so as x goes to negative infinity, the graph will go down into the abyss, and this is negative infinity. Final answer on this one is negative infinity. Here's lots of examples, some easier than others to work with infinity, so good luck with those, keep doing lots of problems and let me know if you have any questions.
