Understanding Limits and Asymptotes

3.5, we're going to talk about limits at infinity. I'll start off with just a little bit of a recall here. Do you remember... I'm sorry. I'm sorry. If the limit as x approaches a number of f of x equal plus or minus infinity. Do you remember that gave us an asymptote? So basically when you could not cross out the problem on your limit, you had an asymptote. When you could cross it out, it's called a removable discontinuity, it's called a hole. This right here will give you an asymptote at the value n. And we had two cases, really. We had this case, where we went to the same infinity, positive infinity on both sides, and said the limit existed. The limit was positive infinity there. Or we had this case. Where one went up, another went down. We said the limit didn't exist there. One's going to positive infinity as we go from the left. One's going to negative infinity as we go from the right. And we had a way to actually determine this, if you remember. How did we determine where we had poles or asymptotes? Scientists would do this, but how do we find out where all this stuff happens? Let's pretend we had a rational function. Rational functions have denominators. How do you find these things? Okay, let me give you an example. Will this have some discontinuities to it? Yes. Absolutely. Where? At 0, 1, 3. Which ones? 0. Is 0 going to be a discontinuity? What do you think? Zero? No. Zero is fine. Zero over a number is okay. That's okay. Where are discontinuities going to exist? Three and negative one. Why? Good, because if a denominator equals zero, you have a problem, right? That's how you find discontinuities. That's why I told you a domain from a long time ago. So what do we do to find discontinuities? Denometer equals zero. That's how we do it. We set denominator equals zero. We solve it down. We're going to get either wholes or asymptotes. Do you remember when we get wholes and when we get asymptotes? Okay, maybe we'll refresh your memory on that next time. So if you'll recall, when we take a limit of a function as we approach some number and it goes to infinity, we're going to have an asymptote. We're going to have either an asymptote where the limit exists as they both go to positive infinity. I didn't draw the case where they both go to negative infinity. where one goes to positive and one goes to negative, and then we'd find that with the sign analysis test. Now, there were two cases where discontinuities existed. We had holes, and we had asymptotes. Do you remember those, the holes and the asymptotes? The holes was what was called a removable discontinuity. What it meant was that only a single point is missing from that function. One little single point. That was what defined a removable discontinuity. And it exists when the numerator and... And the denominator equals 0 at the same time. Why was that the case? Well, if both the numerator and denominator equals 0 at the same point, you're going to have a factor of x minus that value that you could cross out. Remember crossing out those problem areas? That was a removable discontinuity. Now, or a whole. If we don't have that, if the numerator equals a number and the denominator equals zero, and you cannot cross it out and you can't simplify it, it's not removable. That means we have an asymptote. So case two is where we have the asymptote. A vertical asymptote. A vertical asymptote. This is when you can't cancel out. I know you love to use that word cancel out, right? Where you can't cancel out the discontinuity or you can't remove it. Let's do an example here real quick, okay? Let's do an example. Can you tell me where discontinuities exist? So, any discontinuities? How about x equals 0? Is that a discontinuity or not? Is it OK to have 0 in the top of a fraction? Sure, that's 0. That's a point. Give me one discontinuity. 1. 1 would work, sure. Because if I plug in 1, I'm going to get 0 in the denominator. That's a discontinuity. You follow me? So x equals 1 will be a spot in which we will have a discontinuity. Give me another one. Someone left hand side of the room. What do we have? Day 3 would be another one. Sure. Now let me ask you a question about x equals one, x equals negative three. Are they wholes, are they asymptotes? Is one a whole, is one an asymptote? What do you think? And the answer to the question is, can you cross out where the discontinuity happens? So basically, can you cross out Can you cross out the x minus one? Can you cross out the x plus three? So are those wholes or asymptotes? Those are both asymptotes, both of them. Now stop for a second, just watch. What if I had done this please? Which one would be the asymptote in this case? That would be the asymptote here, and that would be a removable discontinuity, or in other words, a hole. Do you follow that? Okay, good. So, I'm going to change it back to this. This is an asymptote. And that's an asymptote. If I have just x over x plus 3 and x minus 1, those are vertical asymptotes. Do you recall how to find out what the limit is around those? Do you remember that? Yep, sine analysis is how you do it. You'd put the negative 3. And you know it's going to be an asymptote because we just talked about that. It's going to be an asymptote there. And you put the positive 1. And you know it's going to be an asymptote. And then all you have to do is plug in some numbers for those intervals. Because look it, look. If it's an asymptote, look at the board here real quick. You know it's going to be one of these cases, right? It's either going to go this, that, or be opposite. So it's going to be one of those. All you have to do is plug in numbers. If it's positive, it's going positive. If it's negative, it's going negative. Same thing in here. Positive, positive, negative, negative. Same thing here. Positive or negative. Now, there's one thing you have to be careful of. One very important thing for us right now. Do you see that the numerator will equal 0 at 0? That point's not giving me an asymptote, but it will change the sign of your fraction. So you're going to put a little marker in here at 0. And the only reason why you're going to do it is because a number in this region could potentially be different than a number in this region concerning the sign. Does that make sense to you? So that's a separator. Because if you just plug in 0, you're going to get 0, right? That doesn't take. tell you positive or negative, that could be a case where you're switching from positive to negative. So you have to put that, plug in a number here and plug in a number here, and that'll tell you. So it's like having four intervals? You have just three intervals, but that's a separating marker which says, I can't plug in just one number here, I need to plug in two different numbers. So you only have two asymptotes. There's not three asymptotes, but it's like, yes, like you're testing four intervals. Okay. Feel okay with this? Why don't we see what happens? Why don't you plug in negative 4? So how about this? U2 rows do negative 4. U2 rows do negative 1. U2 rows do 0.5. And U2 rows do... Okay, can you tell me that? Have you already done the two? When you have them, let me know. It's negative? So you plug it into here, right? You got negative. I don't care about the value, negative. These two rows, have you plugged in negative one? Negative. Negative? Really? Yeah. Negative, positive, negative, positive. How about.5? Positive, positive, negative. How about 2? Here's my question. Will the limit exist, will the limit exist as we approach negative 3? From the left, we're going down. From the right, we're going up. Does it exist? I think no. How about at 1? Nope. From the left, we're going down. From the right, we're going up. If they had been like this, both going up, both going down, sure. How about this question? Trick question. Does the limit exist? Does it exist at zero? Absolutely, that's a point. There's no asymptote there. That's a point. It does exist at zero. Trick job. Thank you, Joe. Feel okay with it so far? Could you do the same thing with something like this? The same thing as something like that. Can you tell me where discontinuities exist on this problem? Where would they exist on this problem? Do you need to distribute all this out to tell that? The only place where discontinuities are going to exist is where the denominator equals 0. True? So where the denominator equals 0 is where this little piece equals 0. You follow? Where is that going to equal zero? Uh-huh. So these are my discontinuities. Would you say that those are holes or asymptotes? Explain why they're asymptotes. Because they can't be removed. Okay, they can't be removed. That's exactly right. Can't be deleted is what she said, but absolutely. Even if you factored that, could you cross it out at all? Definitely not. So these are non-removable discontinuities. Those are asymptotes. So how could you find out where those asymptotes go? Well, this says really what we're concerned about is around the 1. Do you see what I'm talking about? So from our chart here, we've got two discontinuities. We've got a negative one. We've got one. One and negative one. And what we're going to do is check where these asymptotes go. Now, look up here at the board here real quick. Normally, we would check all the asymptotes. Follow? But which one do I really care about? Do I care about the negative one? Why not? Limit says 1, so I don't care about that. I care about this. Sure. Tell me one more point that I absolutely must have up here right now. That 0 could be a change of sign. So where the numerator equals 0, that could also be a place where you want to at least separate. your interval. It's not going to be an asymptote, but you definitely want to plug in, just to be safe, a number between 0 and 1. That's what you want to do. So try that. People on the left-hand side, why don't you try 0.5. Right-hand side, why don't you try 2 for me. See what happens around that. You shouldn't really be able to do it without a calculator, honestly. Because when you think about it, this is going to be a positive number, right? Positive, it doesn't matter, that's positive. Positive over positive is positive. Really? 0.5 is a positive number. You didn't check negative, did you? Because we should be checking between this little interval. That's why we had that zero. 0.5 is positive. This is positive. Positive, positive, positive. Aha! This is the first time I think we've had one of these. Does the limit exist? Yes, it does. This limit, as t approaches 1, That nasty looking function is what can you tell me? positive or negative? So our sign analysis is important our sign analysis. How do you say that? Analysis is? analysis. Oh that's better. Our sign analysis is important for you Are they important? Yeah, you gotta know how to do those. So when you You can cross out, it's very easy, those are holes, no problem. When you can't cross them out, you have to have some way to determine where they're going. And the way you do that is with a sign analysis, just like that. Don't worry about all the asymptotes if I don't ask them for you. Worry about just where the limit says. If I'm asking for all the discontinuities, I'm just asking for the points, right? This was a little extra for us in that case, just wanted to see what happened. This is the application for a limit. I wish I could be okay with this so far. All right, good. The next question we have to answer is what happens with a limit if x doesn't actually go to a number? So for instance, x goes to 1 here, right? Or t in this case. So it very well goes to 1. What if we say I don't want to go to 1, I don't want to go to 2, I want to go to infinity. What happens to our function? I don't want to go to negative 3, I don't want to go to negative 4, I want to go to negative infinity. What happens to our function? That's what this next part is all about. So the question is what happens in a limit or to a function as x approaches positive infinity as far as we can go or negative infinity. Basically x increases forever or decreases forever, what happens to our function. Let's take a look at two examples that are going to give us insight as to this. You know what, let's make a table just to kind of get a picture of what's going on. We'll have x and f. We'll start very easily. We'll start at 1, 10, 100, and then a million. Let's do it for this one. What would happen please if x equal to 1, what would we get? 1. How about if x equal to 10, what would we get? .1..1, very good. Or 110. Yeah, clearly, right? 1 over one of the numbers. How about How about a million? .501..501..501..501..501..501..501..501. Six zeros, right? Five zeros. Five zeros? Yeah, five zeros. What if x equaled a trillion? Would it get bigger or smaller? Smaller. What's it going to? Zero. Is it ever going to reach zero? But think about this. You're dividing a constant number by infinity, right? Like when you're a little kid and you go, you're wrong. Well, you're more wrong. Well, you're wrong plus 10. Well, you're wrong times infinity. Ha, ha, ha, ha, ha. Right? Well, if you did divide it by infinity, you'd have like, you'd have zero. You keep dividing it by a bigger and bigger number, you're going to get zero. Right? One divided by a billion is very close to zero. One divided by a zillion, that's a number, right, number, is very close to zero. But you keep getting bigger and bigger, this quantity is closer and closer to zero. Are you with me on that? Zero. What happens as you go closer to negative infinity? Well, that would be, I'm just going to change this, that would be negative, negative, negative, negative. True. Negative, negative, negative, negative. Is it still approaching zero? Zero. Yes. As we go to negative infinity, we're also approaching zero. What this says is that if f of x, our function, gets really, really close to a number as we approach infinity, then the limit exists. So we can take limits as x approaches one of these infinities if x approaches an actual number. Remember that idea of really close to? That was the idea of that limit. If f of x gets really close to a number as x approaches infinity, plus or minus, whichever one, then the limit exists. Ok, well, if the limit exists, what does it stand for? What's it mean? Think about this. Let's say that you start taking your function and you start going forever in that direction. That's obviously incredible. I've never taken karate. I don't know, but it kind of feels like it. Whatever. The limit move. So, if you ever, by the way, if you ever try to do the math karate, you're going to get your, you're going to beat up. Never do math karate. So you go this way, forever and ever and ever. and your function is going to the same number, is it ever going to get to that number? No. It's getting very, very, very, very close, right? What idea do we have that represents that? But not a vertical one. A vertical one says you're getting really close to that, right, but never reach it. That's the same idea we have going this way. Really close to that, but you're going, we don't have this idea, we only have one, because otherwise we don't have a function. But we have this idea, we're going to get close to a number or close to a number, close to a number or close to a number. Yeah. That's a horizontal asymptote. So now we have two types of asymptotes. Vertical happens when you have discontinuities that are not ruinful. Shake your head or nod your head if you're OK with that. Horizontal happens when you take a limit of your function to infinity. And that's the idea. What does that class look like? By the way, the same rules apply to these limits we're about to do. The rules for limits as x approaches negative and positive infinity are the same as before. So the same rules apply. Everything holds the same. Which means one thing for us. Consider this. What if we take any? any 1 over x to the n power. Do you recall from limits that I can take the limit of a function and then raise it to the power? Do you remember that? That we could pull out the exponent? That was one of our prime rules. That helped us a lot. Well, notice how 1 to the n is always just going to be 1, right? So then this is true. That's a true statement. That's 1 over n. I'm sorry, 1 to the n, that's 1. This is x to the n. That's x to the n. Are you guys okay with that statement? Well, tell me something then. How much is the limit of 1 over x as we approach infinity? What's 0 to the nth power? So what this says in English is that any function where you're dividing a constant, 1, because you can pull a constant out, any constant you can pull out of that limit, any constant by some variable that's... being raised as x approaches to that infinity is going to end up being zero. It's going to have a horizontal asymptote at zero. Zero to the n or zero. Does it make a difference if we're talking about negative infinity? Let's consider that. Let's go to negative infinity. That means, is this still true for negative infinity? Absolutely. Is this still true? Is 1 over x as x approaches negative infinity, is that still zero? Zero to the nth, that holds true for both of them. What this says is that one, a constant over any variable that's being taken to infinity is gonna give you a horizontal asymptote. How many people feel okay with that, with that idea? No matter what the power is, squared, cubed, doesn't matter because you're always taking that one over x to that power. One over x as x approaches infinity, either positive or negative, it's gonna be zero. Zero to any power is still zero. Interesting thing. Horizontal asymptotes in each case at zero. Now before we get into some other computational stuff, let me show you one more idea. Right now we've done a lot of theory so far, so this is not really a whole lot of examples. We're going to change that in just a minute. But I've got to talk about one more. I want you to think of polynomials. Let's talk about the limit of a polynomial. As x approaches positive or negative infinity, think about what you know, think about what you know about polynomials. Would you agree that polynomials are in one of these four cases? Polyphemials are either like this, Or like that. True? Think about x squared. X squared is like this. Does it ever go to an actual number as we go this way? Does it ever go down or up and go to this? No. Polynumerals don't do that. Polynumerals always have a. An x cubed would go this way to that way, right? Negative x cubed goes this way to this way. X squared goes this way. Negative x squared goes this way. They never actually go to an actual number. they're all either going to positive or negative infinity. So as we approach for any polynomial, positive or negative infinity on the x-axis, the function itself approaches positive or negative infinity on the y-axis. Let me say that one more time, because I think I lost some. As you travel this way on the x, your function is doing one of two things. Going this way, forever. Or going this way, forever. You follow me on that one? That's what I just said, okay? If you're going this way on the x-axis, So your function is doing one of two things. Going this way forever, or going this way forever. That's what this is. The limit of a polynomial as x approaches negative infinity or positive infinity goes to positive or negative infinity itself. I'll give you some examples of this. You're going to notice that I always use positive and negative infinity. That's something I do because I don't like to get confused. If I say positive, I write the positive. That's just the way I do it. Do you have to do it that way? No, not necessarily. Helps me. Let's consider this, okay? Now, infinity is not something we can actually plug in. But it's something we can think about. In fact, this one guy actually went crazy thinking about the sizes of infinity because there's different sizes of infinity, which is pretty interesting in itself. So, thinking about infinity is kind of weird to think about, but we can do it. Think about this. Take a big positive number, plug it into X cubed. You know the shape of X cubed, right? It's a polynomial, it's gonna be like this. Take a positive number and plug it in. Do you get a positive or negative? Positive. Take a bigger number. Is it still positive? Take a bigger number. Is it still positive? In fact, it's getting bigger and bigger. This is going to be a positive. I'm going to go to positive infinity. The bigger the positive number I plug in, the bigger the positive number I get out. Let's try this one. Negative infinity. Take a negative number, plug it in, what do you get? Take a smaller negative number, even more negative, plug it in. Do you still get negative? Do you see that this one's actually going to negative infinity? The further we go left on the number line, the further we go down on the y-axis. Negative infinity. And that's what that says. The further you go left, the further your function drops. That's what that's saying. Does that make sense to you? How about this one? X squared. Take a positive number, plug in an x squared. Where are you going? Up or down? Up. Up. Positive infinity. So the bigger the positive number, the bigger the positive number. Try that one though. Take a negative number, plug it in. What do you get? A positive. A positive because x squared takes that number and makes it positive. So even though we're going to the left, our function is going up. That's why x squared looks like that, right? It's going to go up on both sides. This is also positive infinity. Show of hands, how many people feel okay with our limit idea so far? Good, all right, all right. By the way, this is kind of an interesting little note. Did you know that the behavior of the polynomial itself will follow the behavior of the leading term? Did you know that? I'll explain that in a second. I'll explain it in a second. Sorry, I'll say it this way. Limits of a polynomial will follow the behavior of the highest power term. A polynomial, nothing else polynomial. Thank you. I'll give you one quick example because I think you'll really handle this pretty well. Let's say that I want to talk about the limit as x approaches negative infinity of negative 3x cubed minus 2x squared. Minus x. What's 9? No. Here's what you can't do. What you can't do is go, oh, let's see, uh... Negative infinity cubed, negative infinity times negative, that's positive infinity, minus a positive infinity, that's, oh man, I don't know. If you start doing stuff like that, subtracting infinities, that is going to make you go crazy. What you need to understand is that all this subtracting and adding the infinities are meaningless. The behavior of this polynomial is going to follow the term that has the largest power. The term that has the largest power is usually in front, it doesn't have to be, but it's usually in front of your polynomial. It's this one. Here's what the... Look at this little corollary says, what this action theorem says. It says this is going to be equal to the limit as x approaches negative three of just the negative three x cubed. Don't forget the negative, that's important. But the negative three x cubed. You know this intuitively. In fact, when you think about it, look at that. The power is 3. Don't you automatically know it's going to be S-curve? For sure it's going to be doing this. A power of 4 would look like this. 5, 6, 7, 8. You know that, right? Now negatives would be opposite. So negative x cubed, negative of that dude, but no, we don't know what Nancy means. Negative x cubed, negative x to the fourth, negative x to the fifth, negative x to the fifth. It's going to do that, right? You know that, too, at least. It follows that leading term. All the rest of this is garbage as far as the limit's concerned, really, when you talk about infinity. Now, normal limits, no, it's not garbage, but for infinity, that's the one that matters. So consider the leading term, that negative 3x to the third. Take, we're going to negative infinity, mind you. Take and do this. Negative means negative, right? It means a negative number. Take a negative number, plug it in here. What do you get out? Negative number. And then multiply it by negative 3. What do you get, a positive or negative? Negative. So as we're approaching negative infinity, that approaches positive infinity. It's a polynomial. We'll go to one infinity. Let me do this one more time so you really catch us on. You're talking about going to really negative numbers, right? Really negative. Plug in negative numbers to that, you're going to get out negative numbers, right? But then you multiply it by negative 3, which makes them positive numbers. That's positive infinity. Think about negative x cubed. If you think about negative x cubed, it looks like this, doesn't it? About like that. Let's see if I can move my arms a bit. There. It looks about like that. I'm doing dance dance revolution or something. It's this. As you go left, it goes higher, doesn't it? As you go right, it goes low. That's what this is saying. How many people feel okay with that one? I'm not trying to show limits here. Now we're going to start using these ideas to actually compute some limits. This is where kind of the fun comes in. We're back to computing limits. Are they hard? Not really, but you need to know not tricks, but some mathematical manipulations to do them correctly. There's going to be a lot of rationalization, a lot of that. There's going to be a lot of dividing by things. There's a lot of that. Not hard, though. You just got to see it a few times. Hold up that one. Let's see if we can do that. What about that? Where are we going? Positive infinity. Really big numbers. So plug in positive. What's 5 times positive infinity? Infinity. Positive, right? Minus 2? Infinity. What's 3 times infinity? Plus 9. We got you infinity plus one. Ha ha ha ha. So what? Infinity over infinity, one. Yes? No? No, you can't do that. Do you know how big infinity is? Me neither. So how are you supposed to divide it and get one? You can't do that. But maybe we can manipulate this in such a way that this stuff is possible. Now here's the idea. Did you know that you can multiply both the top and the bottom of a fraction by anything that you want? Sure. And if you multiply both the top and the bottom of the fraction by anything you want, it goes to both of those terms, true? What that says is this. What if I divided every term by x? Could I do it? Mathematically, is that legal? Sure. Mathematically, that's fine. 5x minus 2, 3x plus 9. Let's divide all those terms by x. I'll explain where I'm getting the x from in just a minute. Some of you can probably see it. I'll explain why in a bit. You okay with that so far? You sure? Okay, now simplify. What happens here? Same thing happens here, right? Do they cancel out or cross out here? So I get the limit. This becomes 5 minus 2 over x, 3 plus 9 over x. What I show here is how many people can follow that algebra now. Now, watch. Let's take... x to infinity. If I take x to infinity, what happens to the 5? It's 5. Nothing. There's no x. It doesn't change at all. The 5 is 5. What happens to the 3? Nothing. What happens to... to 2 over x? It's really small. Really small. How small? 0. 0. Remember the definition of our limit? Our limit said as we take x approaches infinity, a constant over x is the first thing I gave you today. Really, the first new thing. Said a constant over x to any power goes to what? 0. That was important. We needed that. This becomes how much, folks? Everybody. 0. How much does this become? 0. Constant over infinity. Whether it's positive or negative, it doesn't matter. It's going to 0. So what this says. is this limit, notice how I'm going to stop right at limit now. I've written limit till here. 5 minus 0 over 3 plus 0, that's 5 thirds. Not 1, 5 thirds. Now, is it coincidental? Is it coincidental that, by the way, this works because you do 1 over x over 1 over x. Do you guys see what I'm talking about there? What I really did mathematically was this over that. Do you see it? 1 over x over 1 over x, and I distributed, and that basically... divided every term by x. That's the mathematical way you'd show that. Look at the first expression, the limit of 5x minus 2 over 3x plus 9. What's the leading term here? 5x is the leading term. What's the leading term here? 3. What are the coefficients? 5 and 3. Do you see that when I divide by that x, I'm going to get 5 thirds. Everything else is going to go to 0. So those are polynomials. The leading term is all... that really matters for us for that. Now, do you have to show it? Yeah, I'd like you to show it for right now, okay? As you're going on, you know, later math classes, no problem. But show me that. Where are you getting the x from? It's not the common x here. You're actually only looking at the denominator. And what you're going to do is divide every term by the largest power of x in the denominator. You should probably write that down. Now, why not the numerator? Well, think about it. If your exponents don't exactly match up and you start dividing by the largest power in the numerator, you could have a lot of undefined things. Does that make sense? For instance, if you had this, x, to the fourth over x cubed. And you divide it by x to the fourth, you have other stuff here, but if you divide it by x to the fourth, that's undefined. That's one over x, that's a bad thing. So it's always by the largest power in the denominator. That way you won't be undefined, you might be going to infinity, but you won't be undefined. Does that make sense to you? Another way people like to say this, I'm not going to write it down, it's a shortcut, but if your powers are equal to each other, you're going to the coefficient over the coefficient. That's what we have here. You can see the. that's going to work, right? x squared over x squared, x cubed over x, no problem. If your power is larger on the numerator than the denominator, well, you divide by this power. This is still going to have an x up there somewhere, right? That's infinity, either positive or negative infinity. If this one is larger than, I'm sorry, smaller than this one, if this one's larger, well then you're going to be going to 0. Because you divide by this one, no x's here, x here, you're going to 0. That's one way you can look at that as well. So if the top still has an x, you don't? Of course. Let's try a couple more. You're going to start seeing this very easily. It's not going to be something super duper hard. What's going to be super duper hard is we're going to combine this with some other ideas in a little while. Actually, not even super duper hard then. Super hard, not even a duper, don't worry about the duper. Come on, that was funny. If you were listening to what I was talking about, you could probably tell me where this limit is going right off the bat. Not one-third. If the powers matched up, it would. Where's the power the biggest? That means what we're going to be doing here to show you work, you're going to be dividing everything not by x squared, but by x to the third power. x to the third power, you follow me on that? So this would look just like this. You'd have 5x squared over x to the third minus 4x over x to the third. All over. Everything is divided. Everything. 15x to the third over x to the third minus 3 over x to the third. If we do just a little simplification, well, look what happens. Do you see how this power is still going to be 5 over x? And this is going to be 4 over x squared. You see the x squared? This is going to be 15, sure, but this is going to be 3 over x to the 3rd. We had the principles of beginning class for a reason. It said any constant over any power of x as x approaches any infinity is going where, folks? Where? Zero. So where's this going? Zero. Where's this going? Zero. Where's this going? Okay. Come on, come on, everyone gotta play along. Where's this going? Zero. Zero, zero, 15, zero. So this, after you stop running the limit, would be zero minus zero over 15 minus zero. That's where these things are going. It's a constant over x as x approaches infinity. Negative infinity, but still an infinity. Zero, zero, zero, zero. How much is that gonna be? Zero. That's zero. That says a horizontal asymptote. That's what this is, by the way, a horizontal asymptote. as you're going to the left. Are you following me on that? Now, if you did this, if you reciprocated this, and you divide everything by x squared, do you see how you still have an x on the numerator? That would be saying you're going to positive or negative infinity, depending on what x is approaching. Let's do one more. And then next time we'll start building on this concept. Two more, two more. We've got three minutes now. Again, some of you who kind of understand this concept right now should probably be able to tell me where this is going to approach, or at least have a guess what it's going to approach. Positive and negative. Oh, okay. Positive and negative. Okay. Ooh. Well, you might want to do the work to make sure, but do you see it's not going to go to a constant? Do you see it's not going to go to zero? What are we going to divide by? X cubed, X squared, or X? The largest power in the denominator. So show your work. Don't get lazy on that. Don't take a guess because some of you are guessing right now. Don't take a guess. Show your work. Over x, over x, over x, all over. Now, divide everything by the largest power in the denominator, not the numerator. That'll give you undefined spots. Do you see that? If I divide it by x cubed, I get this whole bunch of undefined stuff. Can't have that. Can't do that. Are you alright with that one so far? Simplify them. You're going to get 7x squared minus 2x plus 1 over x. All over. 9 over x minus 2. You following on that? Now let's kind of think carefully about what happens here. What happens? Where does that go? That goes to positive infinity. Does this stuff matter? No. That goes to positive infinity. Agreed, it's x squared. This is zero, but that is a negative two. Where's that going? Negative infinity. In fact, if you really wanted me to show it to you this way, I could. Couldn't you factor out a negative and get 2 minus 9 over x? Move the negative up top. Negative infinity. Bam. Got it. Did you see it? Did I lose you? You understand I went to positive infinity? Little unorthodox here, but you see this goes to zero. Negative 2 doesn't change. Positive divided by negative is a? Infinity. Infinity divided by 2? Still infinity. Okay. Now mathematically, oh, not that. It doesn't do that. Math. Mathematically, the way that you could show this is the way I showed you. If you factored out the negative, it's negative 2 minus 9 over x. Right? If you take that negative and move it to the top of your fraction, which is legal to do, it goes there. That becomes 2 minus 9 over x. That says this goes to infinity times a negative, that's negative infinity. Divided by a positive now, that's negative infinity. That's what that means as well. Can we show it the way it's on the left and on its way? Yeah. Thank you. Yeah. By the way, did anything with the highest powers of x have anything to do with our problem here? This one and that one is really what caused this to be the way it was. One last thing I'll leave you with, just this idea. Because I want you to look at some of your homework tonight, but I don't want you to be completely afraid of it. Just a little frame. That looks pretty nasty, but tell me something that I can do with exponents. What can you do? Divide by exponents. Not yet. You can move them outside. I can move them outside. Is a root a type of exponent? Then look. You can take a cube root out of the limit. Huh? Now ignore the cube root for a second, just for a second though. Can you do this limit? Yeah. Explain to me how you would do this limit. What would you divide by here? X squared. X, which one? X squared. X squared. Hey, where's this limit going to go from the, on the inside? Two thirds. Do you see the two thirds? Two thirds. Two thirds? And then take a cube root of that. So our answer here, after you show the work, would be a cube root of two thirds. The work you show is this. You show a cube root 2x squared over x squared minus 3 over x squared all over. That's what you show, but that's the answer you get. How many people understood today? I feel okay with it. Right side people, are you guys doing this so far? Yeah? Okay. Okay, so we're talking about how to do limits as x approaches infinity. Now the last example I gave you, I think I gave you that one, didn't I? I said that when you have a cube root, it is a power. That means you can pull it outside of our limit and say, well let's take the cube root of our limit itself. Now when you do that, that's kind of nice, right? Because we know how to take these limits. What we normally do is divide everything by the largest power in the denominator. That's x squared. You would get 2. That's going to be 0. 3. That's going to be 0. You're going to get 2 thirds. You follow? Don't forget about the cube root, but your answer is 2. Two-thirds with a cube root around it. I think that's what I gave you last time. Were you okay with that one? Yeah. Now, I showed you that one again because I want you to consider the problem that's just below it. Can I do the same thing here? For instance, can I pull out... the square root around the whole limit? No. No, because that square root's not around this whole function, it's just on the numerator, so. Oh my gosh, well what do we do? What do we do? Well, let's try to stick with the normal operation of this. When you take an x to infinity, we can't just go, well, that's infinity squared. Okay, that's still infinity. Plus 2 infinity, square root of infinity over infinity. One. We can't do that. But what we did say was maybe we divide by the largest power of x in the denominator. So we're going to look down there. What's the largest power of x in the denominator? x to the first power. So let's divide the numerator and the denominator by x. Are you okay with that so far? Well now here's the issue, and some of you might see this issue already. Can I just take this x inside that square root? I can't unless I have a square root. Remember, you can only combine things if you have the same exact type of root. Follow? Because it's an exponent. You can only put things together if you have the same type of exponent. Well, that's a problem. Does anyone know how to change x into something that I can take into that square root? Let's think about it for a second. Let's think about that x. What do you think? But if I do x squared, I have to change those to x squared. Right? Is that a problem? That would be zero and zero. That would be undefined. So I can't just arbitrarily change x squared, but you're on the right track. 2 over 2 is still 1. x over y, I did 1 over x, I did that. So x squared down here, we can't have an x squared, I need this to still be a value of x. Make it a value of x squared, it's a great idea. x squared times x squared is x to the 4th, that's too much. Say what now? Root of x times root of x is still x. You're close. Oh, you're so close. You're so close. Put that together. You want something that equals x, true? How do you make this equal x? Do that, and they are equal. Is that equal? Okay. No, no, but almost. That's our idea, though. I'm going to fix this in a second. So here's the plan. Why no? that I need this. Do you see why I need that? That is still, well, pretty much x, almost x. And that's going to allow me to take that into that square root. Do you follow me? That's important. But the question is, is x equal to the square root of x squared? And the answer is no. Not really. Because if you think about it, why don't you take like a negative number, negative 5. Is negative 5 equal to the square root of negative 5 squared? No, it's not. Because as soon as you square it, it becomes positive, right? So this is not necessarily true. What is true is this. That's true. So if we have to have this, there's something that's wrong in our problem right now. This is right. What is wrong? Do you see it? You don't see it. Let me go through the process one more time. You knew maybe if I draw in purple, it'll make more sense. You knew if you, you had to do that, right? But you knew that this wouldn't work. So to try to make it equivalent, you did this. That would go into the square root. However, this fails now because this is not equal. That's not equal. Notice you're trying to divide everything by the same thing, right, the same thing. So if you have divided by the same thing, these are not the same thing. This. Is the same thing. That's true. This is true. Yes, this is not true. Not necessarily true. So are the absent value... This is almost true. Is this true? Well, no, almost. This is true. You ready to keep going? Did you understand that idea right now? You sure? Okay, so when we take the square root of x squared, we can't just have x, you have to have the absolute value of x. And I'll show you why it's important in maybe 10 minutes. For right now, let's continue working on this problem. When you have a square root over a square root, that says you can take a square root of the whole entire fraction. That's x squared over x squared plus 2 over x squared. Are you okay on where the x squared over x squared and 2 over x squared are coming from? That's the sole reason why you did that, right? Just to make that in there. Over 3x over the absolute value of x minus 6 over the absolute value of x. Well, that's a limit. X goes to positive infinity. You're going to have the square root of 1 plus 2 over x squared. All of them. Oh, okay. Now we've got to deal with that. I need your minds to think back to what it means to be an absolute value. Absolute value is actually a piecewise function. The piecewise function says, if x is positive, you leave it alone. If x is negative, you change the sign. Does that make sense to you? If x is positive, positive, you leave it alone. If x is negative, you change the sign. And that was the whole idea. That's the piecewise definition of absolute value. Do you remember that? I think we did in here, actually. Well, This is how you apply that. We are going to have the 3x over minus 6 over. Now you get to change your absolute value of x depending on where you are going, whether you go into positive infinity or negative infinity. infinity. Where are we going? Positive numbers or negative numbers? Positive. We're going to positive numbers say that the absolute value of x equals x. Positive numbers say that the absolute value of x equals x. Are you with me on this? The absolute value of x. Now you don't need the absolute value. You know that since I'm going to positive numbers that can become the x. It might seem trivial like well if it's just going to change to x anyway why do we need the absolute value? Trust me, you need them. I'll show you why in a couple minutes, okay? Now I'm like, five minutes. Raise your hand if you feel okay with that so far. You see, now that we have that, those x's do match up. Those x's are gone. You get a limit to positive infinity. One plus two over x squared, all over three. Minus six over x. Tell me some nice things that happen since x is going to positive infinity. Tell me what happens. Where does this go? Not zero. This, the one. The one goes to one. It's a constant. Do constants go anywhere? No. Where does this go? Where does the fraction go? Have we lost it all? Oh gee, you're scaring me, people. You're scaring me here. Limits of constants are constants. Limits of this is not undefined. Where's x going? x is going to infinity. Where's this fraction going? Zero. Zero. A constant over infinity is zero. Divide a constant by infinity. How much are you going to get? Zero. Zero. Think about that. Come on. Where's three going? Three. OK. Where's this going? Zero. Don't forget what we did last time. This says you're going to do. One plus zero over three minus zero. Tell me my final answer, please. What is that? Are you okay with getting the zero and the zero? I think we talked about that last time. A constant over any power of x as x approaches infinity is going to be zero. Because this number is getting huge, and this number is getting huge, and those numbers are staying the same. That means you're going to have a number over huge. A number over huge is zero. Not the square root of 1, okay, 1 third. Show of hands, how many of you feel okay with that? Now, I want you to consider this. Why this is important. Think about it. You can change your problem if you want to. I'm not going to do the whole example over again. I'm not going to give you two examples. I'm going to show you with this. Let's say that that... It is now negative. That means that's negative. Is this still possible? Absolutely. Is this still true? Absolutely. Is that still okay? Yes. At this point, something different would happen. At this point. From here to here. You see, if we're going to negative infinity, that means that for our absolute value of x, where are we going? We're not going here anymore. We're not dealing with positive. numbers we're dealing with negative numbers. That means instead of absolute value of x equaling x, absolute value of x would equal negative x. That means that this and that would be there. Do you see the change? This would be okay. Square, right? Square's going to be positive no matter what. That's where our definition came from. But this would be different. What's 3x over negative x? Minus 6 over negative, well that's a plus. That's not really an effect. that's being plus zero, but this definitely affects us, doesn't it? That's negative three plus zero. That's negative 1 3rd, not positive 1 3rd. What this says is, this is interesting, the first time we've had this, because if you've been thinking about it, you might have said, I said, well, why do we even take positive infinity and negative infinity? Why don't we just do one infinity and consider it a horizontal asymptote the whole way? Well, it doesn't work. Look at that. Our first example was positive infinity, right? That's that way. That was at a horizontal, and remember what these are? A horizontal asymptote? That's a horizontal asymptote at 1 3rd. If you go to the left, negative infinity, is it at 1 3rd? No, it's at negative 1 3rd. So our horizontal asymptotes can be different as you go to positive infinity and negative infinity. That proves it. That's different right there. How many people understood that? Cool. This really is a real thing. Hopefully that made sense to you. We're going to do one more example, maybe two, just to start it off. Let's try that. I'll show you something kind of cool, interesting that you can do. So we give it a try? Looks awesome to me. I don't know about you. Looks awesome to me. Let's try. Ready? What's infinity to the fourth power? Infinity. OK, cool. Plus 2? And the square root of that? And then minus, so infinity, how much is infinity squared? 0. Done. Love that. Woo, easy problems. Yeah? No. Of course, it may be coincidentally, but I really doubt it. We can't do that. You can't just plug in infinities. Whenever you get a case where you're like infinity minus infinity, or infinity over infinity, something like that. that you know something bad's happening. You're not doing something right. So you can't ever do that. You get infinity over infinity. Please, for heaven's sakes, don't put one. Because that would have been one, right? We can't do that. What we do have to do is manipulate the problem somehow. Now in this case, you have a root. You have a root minus something. What's one thing we like to do with roots minus something when we can't work on them any other way? We usually take... to try to rationalize it, all right? Now this doesn't have a denominator, so make one. Put it over what? Put it over what? Get yourself a denominator. Then you're going to try to rationalize. Conjugate. Very good. Conjugate. You want to be fancy about it. Is it plus 2 or minus 2 there, by the way? Plus 2. Good, so this sign doesn't change. This sign does change. And it's over exactly the same thing. Are you okay with that so far? Yes or no? Yes. Well, the denominator has nothing to distribute, which is really nice. We don't have to do any work up there. The numerator, just distribute it carefully. You know with conjugates, your middle term should be bye-bye, right? Your middle term should be gone. Also, the conjugate works this way because the square root times the square root gives you the radicand that's inside. That's nice. So our limit to positive infinity becomes something over the square root of x to the fourth plus two plus x squared. The numerator becomes, oh, what's the numerator become? Distribute, right? Do you see that on your distribution? Because you should have those. X to the fourth plus two. We're going to have plus and minus, the same exact thing. And then we're going to get, what's the last thing? You see the minus x to the fourth of 2? Don't forget about the exponent. If you make the exponent incorrect, it's not going to come out right. So you really need to be careful about your distribution. Don't forget to distribute that last part. You got me? What happens here that's nice? Yeah, oh that's great. I love that. 2 square root of... I'm going to go ahead and do that. x to the fourth plus two plus x squared. So far so good? Okay. Can we do that legally? Can we just say two over a fifth? Well, I don't know. Especially if this is a, especially if you did this. Especially if you did that, that would be a big problem. Do you see that? That would be an issue. So what we're going to do is make sure about it. You're going to start dividing by the largest power in the denominator. Again, just like you normally would. So what's the largest power, especially if I give you this one, okay? What if that was 2x? That would be a big problem. This one, yeah, Scott's right. You probably could look at this and go, that's 2. That's infinity. That's infinity. There's no way that that's not going to be infinity. You got me? this is going to go to zero. He's right. But to check your work, especially if you have something like this, that might not go to infinity. That definitely wouldn't go to infinity. Okay, that wouldn't. So to check your work to show that, what you're going to do is go, all right, well, let's divide everything by the largest power in the denominator. Now, don't trick yourself here. The largest power in the denominator is not x to the fourth. It's not x to the fourth. What is it? The way you can think about it. Cover up the rest of it. of the largest term, what's the square root of x to the fourth? It's x squared. That's your largest power. It's the square root of that because it's inside the square root. So divide everything by that. I'm making you do this because I want you to see one more time this setup. Notice we're divided by x squared. If I divide this by x squared, is that going to work here? Is this okay? No. Can I take that inside of that square root? Okay, so think back to what we did. This is fine, right? This is fine. That's great. That's what we like. What does this do? What now? Oh, okay. We got stuttered again. Uh-oh. Square root of x to the fourth would be great. Did you say that already? I was thinking, my mind was wandering. I was thinking about the Simpsons. So, well since this is x squared and this is x squared, We need to make this equal to x squared, but it also has to have a square root around it. You follow? You can't just do this because that's x. So change the power. Is that true? Now, some of you are going to ask me, well, wait a second Mr. Leonard, don't you have to have some absolute value here? Let me ask you this question. Is this always true? That actually is always true, no matter what. No absolute value needed because you have all positive, all positive. It's always going to be the same. Does that make sense? So we don't even need that for this case. So here we're going to have, I just wanted you to see that one time, 2 over x squared, all over, we're going to have the square root. If you think about it, this is going to be 1. Plus 2 over x to the fourth. Did you see the 1 plus 2 over x to the fourth? Yes, no? We get the 1x to the fourth over x to the fourth plus the 2 over x to the fourth. That was one. Show of hands, how many people feel okay getting down to that far? Oh, that's not many. Let's have some questions if you guys are not so okay on this. Are you okay on rationalizing? Yeah? Are you okay on crossing those things out? That should be easy. That's the fun part. Are you okay on dividing by x squared because that is the largest power in your denominator? Are you okay on doing the square root of x to the fourth because that's still x squared? Then we take it inside, we have x to the fourth over x to the fourth, that's one. Two over x to the fourth, two over x to the fourth. x squared over x squared, that's one. Where does this go, ladies and gentlemen, when we're taking it to infinity? Where does that go? Is it okay to have zero on the top of the fraction? Where does this go? Where does this go? Where does this go? Would you agree that this denominator goes to two? So this equals... 0 over 2 or 0. You're going to notice in this case, if I change that to a negative, nothing about this changes. I don't have any absolute value. Nothing changes. That says the horizontal asymptote in both directions is at 0. That's what it's going to be. You following me on that one? Would you like to see one more case? Yeah. Would that be all positive? ...experience? Or even experience? Would that be the case? We just started with the evil one there. That was...I'm sorry, I'll go there. That was evil. Oh, I don't know. It really depends on what you have. I can't say a cover-all situation for every time. Should we change the problem just a little bit? Let's change the problem just a little bit, alright? So what I'm going to do, instead of doing a whole completely new one, do you have any questions on this? We're going to see how it changes if I do this little bit to it. That. If that changes our problem at all. Let's see the things that change. Let's go through our problem. I'll try to use a different colored pen so you see the differences. Can you still rationalize? What's the only thing that changes about our rationalization? That's going to be an x squared. Are you all right with that so far? What changes here, ladies and gentlemen, when I distribute? Just an x squared. You see it? It would be kind of silly to do the whole example again, right? It's pretty much the same thing. That would be an x squared, and this would be an x squared. You still okay? Well, what happens here then is this is an x squared, and that's an x squared. True? Okay. So, okay, well, hang on a second then. Well, if that's the case, Are we still going to be dividing by x squared down here? Yeah, because that's still the largest power. That's not. That actually, if you looked at it, would be x. The square root of x squared is x, so that wouldn't be the largest power. We still divide by x squared, but now I have an x squared right there. That 2x squared, that's right there. This would look almost the same as if I have an x squared there. What that means is I'll have a limit as x approaches positive infinity of 2 over, this is going to give you 1, plus 2 over x squared. You know I'll show all the work for you in case you're having trouble following me. X to the fourth over x to the fourth plus 2x squared over x to the fourth, x squared over x squared. Are you guys satisfied with that one? Hopefully? Yeah. Okay, good. This gives you 1, so we're still at 2. That gives you 1. This gives you 2 over x squared, and that gives you 1. We're now ready to take the limit. Ready to let the x go to infinity. What happens to our 2? Does it go to 0? It didn't last time because we had an over x. Where does that go? Where does that go? Where does that go? What is it? We've got a 2 over the square root of 1 plus 0 plus 1 plus 0. 1 plus 0 plus 1. That's the square root of 1, that's 1. 1 plus 1 is 2. We get 2 over 2. Did it change? So, a little bit. So that little change there said we're not at 0 anymore, we have a horizontal asymptote at now 1. So we're going at 1 when we get to infinity. How do you feel okay with what we've talked about so far? This is such good stuff. Just kidding. I'm just blowing out. Allergies are getting to me. Okay, last one. Is it making sense so far? Can you follow that? Yes. Shane, you okay? Yeah. All right. Oh, that's nasty. Still sure you're okay with that one? How about that one? A little tough? Conjugate. There's nothing to sing, just conjugate. Here. There's nothing to distribute. On the top and the bottom, you're... Just think about it for a second. No derivatives. Oh, good try though. When you don't know the answer, just derivative that. Good try. This is one of those 5% Oh, man. Yeah, close. Huh? Why? It's going to, you have a negative under the square root. Ah. So if I take really big positive numbers, subtract them from 7, are they going to be positive or negative? And they're under a square root. Doesn't exist. That was the easiest problem of the day. Come on. No, it doesn't exist. So think about your limits. I mean, you can think of them like that. You have a 7 minus something huge. 7 minus something huge is something really negative. Square root of something negative is undefined. The full limit does not exist. Does that make sense? Now, if I did this... It's not harder. It's the same idea. You have 7 plus a really big number. The 7 plus a really big number squared over that is still a really big number. It's infinity. Alright, so you can think of that. The only problems you really come up with is when you do things like infinity plus or minus infinity. infinity, infinity over infinity, that sort of thing gets you. You can't do those. But if you know it's going to infinity or it's not going anywhere, that's fine. That's okay. You can consider things like that. There's no denominator, nothing to divide by, you're good. Does that make sense? So negative infinity would still be? This, no, this is actually. Positive. Yes. Do you feel okay with our limits? Have I blown your minds yet today? So, when you have issues where you're subtracting infinities, rationalize them. If you have a square root, rationalize them. If you don't think through the limit what it means to go to infinity, consider the type of problems you have. You already know polynomials, any polynomial is going to go to positive or negative infinity, right? We talked about that last time. Said, well, it follows really the leading coefficient, or leading term, sorry, the power of leading term. So, x cubed is going to go up, or if it's negative x cubed, it's going to go down, and things like that happen. So, be prepared for those limits. Are there any questions before we continue? All right.

3.5, we're going to talk about limits at infinity. I'll start off with just a little bit of a recall here. Do you remember...

I'm sorry. I'm sorry. If the limit as x approaches a number of f of x equal plus or minus infinity.

Do you remember that gave us an asymptote? So basically when you could not cross out the problem on your limit, you had an asymptote. When you could cross it out, it's called a removable discontinuity, it's called a hole. This right here will give you an asymptote at the value n. And we had two cases, really.

We had this case, where we went to the same infinity, positive infinity on both sides, and said the limit existed. The limit was positive infinity there. Or we had this case.

Where one went up, another went down. We said the limit didn't exist there. One's going to positive infinity as we go from the left.

One's going to negative infinity as we go from the right. And we had a way to actually determine this, if you remember. How did we determine where we had poles or asymptotes? Scientists would do this, but how do we find out where all this stuff happens?

Let's pretend we had a rational function. Rational functions have denominators. How do you find these things? Okay, let me give you an example. Will this have some discontinuities to it?

Yes. Absolutely. Where? At 0, 1, 3. Which ones? 0. Is 0 going to be a discontinuity?

What do you think? Zero? No.

Zero is fine. Zero over a number is okay. That's okay. Where are discontinuities going to exist? Three and negative one.

Why? Good, because if a denominator equals zero, you have a problem, right? That's how you find discontinuities. That's why I told you a domain from a long time ago. So what do we do to find discontinuities?

Denometer equals zero. That's how we do it. We set denominator equals zero.

We solve it down. We're going to get either wholes or asymptotes. Do you remember when we get wholes and when we get asymptotes? Okay, maybe we'll refresh your memory on that next time.

So if you'll recall, when we take a limit of a function as we approach some number and it goes to infinity, we're going to have an asymptote. We're going to have either an asymptote where the limit exists as they both go to positive infinity. I didn't draw the case where they both go to negative infinity. where one goes to positive and one goes to negative, and then we'd find that with the sign analysis test.

Now, there were two cases where discontinuities existed. We had holes, and we had asymptotes. Do you remember those, the holes and the asymptotes? The holes was what was called a removable discontinuity.

What it meant was that only a single point is missing from that function. One little single point. That was what defined a removable discontinuity.

And it exists when the numerator and... And the denominator equals 0 at the same time. Why was that the case? Well, if both the numerator and denominator equals 0 at the same point, you're going to have a factor of x minus that value that you could cross out.

Remember crossing out those problem areas? That was a removable discontinuity. Now, or a whole.

If we don't have that, if the numerator equals a number and the denominator equals zero, and you cannot cross it out and you can't simplify it, it's not removable. That means we have an asymptote. So case two is where we have the asymptote.

A vertical asymptote. A vertical asymptote. This is when you can't cancel out.

I know you love to use that word cancel out, right? Where you can't cancel out the discontinuity or you can't remove it. Let's do an example here real quick, okay? Let's do an example. Can you tell me where discontinuities exist?

So, any discontinuities? How about x equals 0? Is that a discontinuity or not? Is it OK to have 0 in the top of a fraction?

Sure, that's 0. That's a point. Give me one discontinuity. 1. 1 would work, sure. Because if I plug in 1, I'm going to get 0 in the denominator.

That's a discontinuity. You follow me? So x equals 1 will be a spot in which we will have a discontinuity.

Give me another one. Someone left hand side of the room. What do we have? Day 3 would be another one.

Sure. Now let me ask you a question about x equals one, x equals negative three. Are they wholes, are they asymptotes? Is one a whole, is one an asymptote?

What do you think? And the answer to the question is, can you cross out where the discontinuity happens? So basically, can you cross out Can you cross out the x minus one? Can you cross out the x plus three?

So are those wholes or asymptotes? Those are both asymptotes, both of them. Now stop for a second, just watch.

What if I had done this please? Which one would be the asymptote in this case? That would be the asymptote here, and that would be a removable discontinuity, or in other words, a hole.

Do you follow that? Okay, good. So, I'm going to change it back to this. This is an asymptote.

And that's an asymptote. If I have just x over x plus 3 and x minus 1, those are vertical asymptotes. Do you recall how to find out what the limit is around those? Do you remember that? Yep, sine analysis is how you do it.

You'd put the negative 3. And you know it's going to be an asymptote because we just talked about that. It's going to be an asymptote there. And you put the positive 1. And you know it's going to be an asymptote.

And then all you have to do is plug in some numbers for those intervals. Because look it, look. If it's an asymptote, look at the board here real quick. You know it's going to be one of these cases, right? It's either going to go this, that, or be opposite.

So it's going to be one of those. All you have to do is plug in numbers. If it's positive, it's going positive.

If it's negative, it's going negative. Same thing in here. Positive, positive, negative, negative.

Same thing here. Positive or negative. Now, there's one thing you have to be careful of.

One very important thing for us right now. Do you see that the numerator will equal 0 at 0? That point's not giving me an asymptote, but it will change the sign of your fraction. So you're going to put a little marker in here at 0. And the only reason why you're going to do it is because a number in this region could potentially be different than a number in this region concerning the sign.

Does that make sense to you? So that's a separator. Because if you just plug in 0, you're going to get 0, right? That doesn't take. tell you positive or negative, that could be a case where you're switching from positive to negative.

So you have to put that, plug in a number here and plug in a number here, and that'll tell you. So it's like having four intervals? You have just three intervals, but that's a separating marker which says, I can't plug in just one number here, I need to plug in two different numbers. So you only have two asymptotes.

There's not three asymptotes, but it's like, yes, like you're testing four intervals. Okay. Feel okay with this? Why don't we see what happens? Why don't you plug in negative 4?

So how about this? U2 rows do negative 4. U2 rows do negative 1. U2 rows do 0.5. And U2 rows do... Okay, can you tell me that?

Have you already done the two? When you have them, let me know. It's negative? So you plug it into here, right?

You got negative. I don't care about the value, negative. These two rows, have you plugged in negative one? Negative.

Negative? Really? Yeah.

Negative, positive, negative, positive. How about.5? Positive, positive, negative. How about 2?

Here's my question. Will the limit exist, will the limit exist as we approach negative 3? From the left, we're going down.

From the right, we're going up. Does it exist? I think no. How about at 1?

Nope. From the left, we're going down. From the right, we're going up. If they had been like this, both going up, both going down, sure. How about this question?

Trick question. Does the limit exist? Does it exist at zero?

Absolutely, that's a point. There's no asymptote there. That's a point.

It does exist at zero. Trick job. Thank you, Joe. Feel okay with it so far? Could you do the same thing with something like this?

The same thing as something like that. Can you tell me where discontinuities exist on this problem? Where would they exist on this problem? Do you need to distribute all this out to tell that?

The only place where discontinuities are going to exist is where the denominator equals 0. True? So where the denominator equals 0 is where this little piece equals 0. You follow? Where is that going to equal zero? Uh-huh.

So these are my discontinuities. Would you say that those are holes or asymptotes? Explain why they're asymptotes.

Because they can't be removed. Okay, they can't be removed. That's exactly right.

Can't be deleted is what she said, but absolutely. Even if you factored that, could you cross it out at all? Definitely not. So these are non-removable discontinuities. Those are asymptotes.

So how could you find out where those asymptotes go? Well, this says really what we're concerned about is around the 1. Do you see what I'm talking about? So from our chart here, we've got two discontinuities.

We've got a negative one. We've got one. One and negative one. And what we're going to do is check where these asymptotes go.

Now, look up here at the board here real quick. Normally, we would check all the asymptotes. Follow?

But which one do I really care about? Do I care about the negative one? Why not? Limit says 1, so I don't care about that.

I care about this. Sure. Tell me one more point that I absolutely must have up here right now.

That 0 could be a change of sign. So where the numerator equals 0, that could also be a place where you want to at least separate. your interval. It's not going to be an asymptote, but you definitely want to plug in, just to be safe, a number between 0 and 1. That's what you want to do.

So try that. People on the left-hand side, why don't you try 0.5. Right-hand side, why don't you try 2 for me.

See what happens around that. You shouldn't really be able to do it without a calculator, honestly. Because when you think about it, this is going to be a positive number, right? Positive, it doesn't matter, that's positive.

Positive over positive is positive. Really? 0.5 is a positive number.

You didn't check negative, did you? Because we should be checking between this little interval. That's why we had that zero. 0.5 is positive. This is positive.

Positive, positive, positive. Aha! This is the first time I think we've had one of these. Does the limit exist?

Yes, it does. This limit, as t approaches 1, That nasty looking function is what can you tell me? positive or negative?

So our sign analysis is important our sign analysis. How do you say that? Analysis is? analysis.

Oh that's better. Our sign analysis is important for you Are they important? Yeah, you gotta know how to do those. So when you You can cross out, it's very easy, those are holes, no problem.

When you can't cross them out, you have to have some way to determine where they're going. And the way you do that is with a sign analysis, just like that. Don't worry about all the asymptotes if I don't ask them for you.

Worry about just where the limit says. If I'm asking for all the discontinuities, I'm just asking for the points, right? This was a little extra for us in that case, just wanted to see what happened.

This is the application for a limit. I wish I could be okay with this so far. All right, good.

The next question we have to answer is what happens with a limit if x doesn't actually go to a number? So for instance, x goes to 1 here, right? Or t in this case.

So it very well goes to 1. What if we say I don't want to go to 1, I don't want to go to 2, I want to go to infinity. What happens to our function? I don't want to go to negative 3, I don't want to go to negative 4, I want to go to negative infinity. What happens to our function? That's what this next part is all about.

So the question is what happens in a limit or to a function as x approaches positive infinity as far as we can go or negative infinity. Basically x increases forever or decreases forever, what happens to our function. Let's take a look at two examples that are going to give us insight as to this.

You know what, let's make a table just to kind of get a picture of what's going on. We'll have x and f. We'll start very easily.

We'll start at 1, 10, 100, and then a million. Let's do it for this one. What would happen please if x equal to 1, what would we get?

1. How about if x equal to 10, what would we get? .1..1, very good. Or 110. Yeah, clearly, right? 1 over one of the numbers. How about How about a million?

.501..501..501..501..501..501..501..501. Six zeros, right? Five zeros.

Five zeros? Yeah, five zeros. What if x equaled a trillion? Would it get bigger or smaller?

Smaller. What's it going to? Zero. Is it ever going to reach zero?

But think about this. You're dividing a constant number by infinity, right? Like when you're a little kid and you go, you're wrong.

Well, you're more wrong. Well, you're wrong plus 10. Well, you're wrong times infinity. Ha, ha, ha, ha, ha. Right?

Well, if you did divide it by infinity, you'd have like, you'd have zero. You keep dividing it by a bigger and bigger number, you're going to get zero. Right? One divided by a billion is very close to zero. One divided by a zillion, that's a number, right, number, is very close to zero.

But you keep getting bigger and bigger, this quantity is closer and closer to zero. Are you with me on that? Zero. What happens as you go closer to negative infinity? Well, that would be, I'm just going to change this, that would be negative, negative, negative, negative.

True. Negative, negative, negative, negative. Is it still approaching zero? Zero.

Yes. As we go to negative infinity, we're also approaching zero. What this says is that if f of x, our function, gets really, really close to a number as we approach infinity, then the limit exists. So we can take limits as x approaches one of these infinities if x approaches an actual number. Remember that idea of really close to?

That was the idea of that limit. If f of x gets really close to a number as x approaches infinity, plus or minus, whichever one, then the limit exists. Ok, well, if the limit exists, what does it stand for?

What's it mean? Think about this. Let's say that you start taking your function and you start going forever in that direction.

That's obviously incredible. I've never taken karate. I don't know, but it kind of feels like it. Whatever. The limit move.

So, if you ever, by the way, if you ever try to do the math karate, you're going to get your, you're going to beat up. Never do math karate. So you go this way, forever and ever and ever.

and your function is going to the same number, is it ever going to get to that number? No. It's getting very, very, very, very close, right?

What idea do we have that represents that? But not a vertical one. A vertical one says you're getting really close to that, right, but never reach it. That's the same idea we have going this way.

Really close to that, but you're going, we don't have this idea, we only have one, because otherwise we don't have a function. But we have this idea, we're going to get close to a number or close to a number, close to a number or close to a number. Yeah.

That's a horizontal asymptote. So now we have two types of asymptotes. Vertical happens when you have discontinuities that are not ruinful. Shake your head or nod your head if you're OK with that.

Horizontal happens when you take a limit of your function to infinity. And that's the idea. What does that class look like?

By the way, the same rules apply to these limits we're about to do. The rules for limits as x approaches negative and positive infinity are the same as before. So the same rules apply.

Everything holds the same. Which means one thing for us. Consider this.

What if we take any? any 1 over x to the n power. Do you recall from limits that I can take the limit of a function and then raise it to the power?

Do you remember that? That we could pull out the exponent? That was one of our prime rules. That helped us a lot.

Well, notice how 1 to the n is always just going to be 1, right? So then this is true. That's a true statement.

That's 1 over n. I'm sorry, 1 to the n, that's 1. This is x to the n. That's x to the n. Are you guys okay with that statement?

Well, tell me something then. How much is the limit of 1 over x as we approach infinity? What's 0 to the nth power?

So what this says in English is that any function where you're dividing a constant, 1, because you can pull a constant out, any constant you can pull out of that limit, any constant by some variable that's... being raised as x approaches to that infinity is going to end up being zero. It's going to have a horizontal asymptote at zero.

Zero to the n or zero. Does it make a difference if we're talking about negative infinity? Let's consider that. Let's go to negative infinity. That means, is this still true for negative infinity?

Absolutely. Is this still true? Is 1 over x as x approaches negative infinity, is that still zero? Zero to the nth, that holds true for both of them. What this says is that one, a constant over any variable that's being taken to infinity is gonna give you a horizontal asymptote.

How many people feel okay with that, with that idea? No matter what the power is, squared, cubed, doesn't matter because you're always taking that one over x to that power. One over x as x approaches infinity, either positive or negative, it's gonna be zero. Zero to any power is still zero. Interesting thing.

Horizontal asymptotes in each case at zero. Now before we get into some other computational stuff, let me show you one more idea. Right now we've done a lot of theory so far, so this is not really a whole lot of examples.

We're going to change that in just a minute. But I've got to talk about one more. I want you to think of polynomials. Let's talk about the limit of a polynomial.

As x approaches positive or negative infinity, think about what you know, think about what you know about polynomials. Would you agree that polynomials are in one of these four cases? Polyphemials are either like this, Or like that. True? Think about x squared.

X squared is like this. Does it ever go to an actual number as we go this way? Does it ever go down or up and go to this? No.

Polynumerals don't do that. Polynumerals always have a. An x cubed would go this way to that way, right? Negative x cubed goes this way to this way. X squared goes this way.

Negative x squared goes this way. They never actually go to an actual number. they're all either going to positive or negative infinity. So as we approach for any polynomial, positive or negative infinity on the x-axis, the function itself approaches positive or negative infinity on the y-axis.

Let me say that one more time, because I think I lost some. As you travel this way on the x, your function is doing one of two things. Going this way, forever. Or going this way, forever.

You follow me on that one? That's what I just said, okay? If you're going this way on the x-axis, So your function is doing one of two things. Going this way forever, or going this way forever.

That's what this is. The limit of a polynomial as x approaches negative infinity or positive infinity goes to positive or negative infinity itself. I'll give you some examples of this. You're going to notice that I always use positive and negative infinity. That's something I do because I don't like to get confused.

If I say positive, I write the positive. That's just the way I do it. Do you have to do it that way?

No, not necessarily. Helps me. Let's consider this, okay? Now, infinity is not something we can actually plug in. But it's something we can think about.

In fact, this one guy actually went crazy thinking about the sizes of infinity because there's different sizes of infinity, which is pretty interesting in itself. So, thinking about infinity is kind of weird to think about, but we can do it. Think about this. Take a big positive number, plug it into X cubed.

You know the shape of X cubed, right? It's a polynomial, it's gonna be like this. Take a positive number and plug it in. Do you get a positive or negative?

Positive. Take a bigger number. Is it still positive?

Take a bigger number. Is it still positive? In fact, it's getting bigger and bigger. This is going to be a positive.

I'm going to go to positive infinity. The bigger the positive number I plug in, the bigger the positive number I get out. Let's try this one.

Negative infinity. Take a negative number, plug it in, what do you get? Take a smaller negative number, even more negative, plug it in. Do you still get negative?

Do you see that this one's actually going to negative infinity? The further we go left on the number line, the further we go down on the y-axis. Negative infinity.

And that's what that says. The further you go left, the further your function drops. That's what that's saying.

Does that make sense to you? How about this one? X squared. Take a positive number, plug in an x squared. Where are you going?

Up or down? Up. Up. Positive infinity.

So the bigger the positive number, the bigger the positive number. Try that one though. Take a negative number, plug it in.

What do you get? A positive. A positive because x squared takes that number and makes it positive. So even though we're going to the left, our function is going up. That's why x squared looks like that, right?

It's going to go up on both sides. This is also positive infinity. Show of hands, how many people feel okay with our limit idea so far?

Good, all right, all right. By the way, this is kind of an interesting little note. Did you know that the behavior of the polynomial itself will follow the behavior of the leading term? Did you know that?

I'll explain that in a second. I'll explain it in a second. Sorry, I'll say it this way. Limits of a polynomial will follow the behavior of the highest power term. A polynomial, nothing else polynomial.

Thank you. I'll give you one quick example because I think you'll really handle this pretty well. Let's say that I want to talk about the limit as x approaches negative infinity of negative 3x cubed minus 2x squared.

Minus x. What's 9? No. Here's what you can't do.

What you can't do is go, oh, let's see, uh... Negative infinity cubed, negative infinity times negative, that's positive infinity, minus a positive infinity, that's, oh man, I don't know. If you start doing stuff like that, subtracting infinities, that is going to make you go crazy.

What you need to understand is that all this subtracting and adding the infinities are meaningless. The behavior of this polynomial is going to follow the term that has the largest power. The term that has the largest power is usually in front, it doesn't have to be, but it's usually in front of your polynomial. It's this one. Here's what the...

Look at this little corollary says, what this action theorem says. It says this is going to be equal to the limit as x approaches negative three of just the negative three x cubed. Don't forget the negative, that's important.

But the negative three x cubed. You know this intuitively. In fact, when you think about it, look at that. The power is 3. Don't you automatically know it's going to be S-curve? For sure it's going to be doing this.

A power of 4 would look like this. 5, 6, 7, 8. You know that, right? Now negatives would be opposite. So negative x cubed, negative of that dude, but no, we don't know what Nancy means.

Negative x cubed, negative x to the fourth, negative x to the fifth, negative x to the fifth. It's going to do that, right? You know that, too, at least.

It follows that leading term. All the rest of this is garbage as far as the limit's concerned, really, when you talk about infinity. Now, normal limits, no, it's not garbage, but for infinity, that's the one that matters. So consider the leading term, that negative 3x to the third. Take, we're going to negative infinity, mind you.

Take and do this. Negative means negative, right? It means a negative number. Take a negative number, plug it in here. What do you get out?

Negative number. And then multiply it by negative 3. What do you get, a positive or negative? Negative.

So as we're approaching negative infinity, that approaches positive infinity. It's a polynomial. We'll go to one infinity. Let me do this one more time so you really catch us on.

You're talking about going to really negative numbers, right? Really negative. Plug in negative numbers to that, you're going to get out negative numbers, right?

But then you multiply it by negative 3, which makes them positive numbers. That's positive infinity. Think about negative x cubed. If you think about negative x cubed, it looks like this, doesn't it? About like that.

Let's see if I can move my arms a bit. There. It looks about like that.

I'm doing dance dance revolution or something. It's this. As you go left, it goes higher, doesn't it?

As you go right, it goes low. That's what this is saying. How many people feel okay with that one? I'm not trying to show limits here. Now we're going to start using these ideas to actually compute some limits.

This is where kind of the fun comes in. We're back to computing limits. Are they hard? Not really, but you need to know not tricks, but some mathematical manipulations to do them correctly. There's going to be a lot of rationalization, a lot of that.

There's going to be a lot of dividing by things. There's a lot of that. Not hard, though. You just got to see it a few times.

Hold up that one. Let's see if we can do that. What about that? Where are we going? Positive infinity.

Really big numbers. So plug in positive. What's 5 times positive infinity?

Infinity. Positive, right? Minus 2?

Infinity. What's 3 times infinity? Plus 9. We got you infinity plus one. Ha ha ha ha. So what?

Infinity over infinity, one. Yes? No? No, you can't do that. Do you know how big infinity is?

Me neither. So how are you supposed to divide it and get one? You can't do that. But maybe we can manipulate this in such a way that this stuff is possible.

Now here's the idea. Did you know that you can multiply both the top and the bottom of a fraction by anything that you want? Sure. And if you multiply both the top and the bottom of the fraction by anything you want, it goes to both of those terms, true? What that says is this.

What if I divided every term by x? Could I do it? Mathematically, is that legal?

Sure. Mathematically, that's fine. 5x minus 2, 3x plus 9. Let's divide all those terms by x.

I'll explain where I'm getting the x from in just a minute. Some of you can probably see it. I'll explain why in a bit.

You okay with that so far? You sure? Okay, now simplify. What happens here? Same thing happens here, right?

Do they cancel out or cross out here? So I get the limit. This becomes 5 minus 2 over x, 3 plus 9 over x. What I show here is how many people can follow that algebra now.

Now, watch. Let's take... x to infinity. If I take x to infinity, what happens to the 5?

It's 5. Nothing. There's no x. It doesn't change at all. The 5 is 5. What happens to the 3? Nothing.

What happens to... to 2 over x? It's really small.

Really small. How small? 0. 0. Remember the definition of our limit?

Our limit said as we take x approaches infinity, a constant over x is the first thing I gave you today. Really, the first new thing. Said a constant over x to any power goes to what? 0. That was important. We needed that.

This becomes how much, folks? Everybody. 0. How much does this become?

0. Constant over infinity. Whether it's positive or negative, it doesn't matter. It's going to 0. So what this says.

is this limit, notice how I'm going to stop right at limit now. I've written limit till here. 5 minus 0 over 3 plus 0, that's 5 thirds.

Not 1, 5 thirds. Now, is it coincidental? Is it coincidental that, by the way, this works because you do 1 over x over 1 over x. Do you guys see what I'm talking about there? What I really did mathematically was this over that.

Do you see it? 1 over x over 1 over x, and I distributed, and that basically... divided every term by x. That's the mathematical way you'd show that.

Look at the first expression, the limit of 5x minus 2 over 3x plus 9. What's the leading term here? 5x is the leading term. What's the leading term here?

3. What are the coefficients? 5 and 3. Do you see that when I divide by that x, I'm going to get 5 thirds. Everything else is going to go to 0. So those are polynomials.

The leading term is all... that really matters for us for that. Now, do you have to show it? Yeah, I'd like you to show it for right now, okay? As you're going on, you know, later math classes, no problem.

But show me that. Where are you getting the x from? It's not the common x here.

You're actually only looking at the denominator. And what you're going to do is divide every term by the largest power of x in the denominator. You should probably write that down.

Now, why not the numerator? Well, think about it. If your exponents don't exactly match up and you start dividing by the largest power in the numerator, you could have a lot of undefined things.

Does that make sense? For instance, if you had this, x, to the fourth over x cubed. And you divide it by x to the fourth, you have other stuff here, but if you divide it by x to the fourth, that's undefined. That's one over x, that's a bad thing.

So it's always by the largest power in the denominator. That way you won't be undefined, you might be going to infinity, but you won't be undefined. Does that make sense to you? Another way people like to say this, I'm not going to write it down, it's a shortcut, but if your powers are equal to each other, you're going to the coefficient over the coefficient.

That's what we have here. You can see the. that's going to work, right? x squared over x squared, x cubed over x, no problem.

If your power is larger on the numerator than the denominator, well, you divide by this power. This is still going to have an x up there somewhere, right? That's infinity, either positive or negative infinity.

If this one is larger than, I'm sorry, smaller than this one, if this one's larger, well then you're going to be going to 0. Because you divide by this one, no x's here, x here, you're going to 0. That's one way you can look at that as well. So if the top still has an x, you don't? Of course.

Let's try a couple more. You're going to start seeing this very easily. It's not going to be something super duper hard.

What's going to be super duper hard is we're going to combine this with some other ideas in a little while. Actually, not even super duper hard then. Super hard, not even a duper, don't worry about the duper. Come on, that was funny.

If you were listening to what I was talking about, you could probably tell me where this limit is going right off the bat. Not one-third. If the powers matched up, it would. Where's the power the biggest? That means what we're going to be doing here to show you work, you're going to be dividing everything not by x squared, but by x to the third power.

x to the third power, you follow me on that? So this would look just like this. You'd have 5x squared over x to the third minus 4x over x to the third. All over.

Everything is divided. Everything. 15x to the third over x to the third minus 3 over x to the third. If we do just a little simplification, well, look what happens. Do you see how this power is still going to be 5 over x?

And this is going to be 4 over x squared. You see the x squared? This is going to be 15, sure, but this is going to be 3 over x to the 3rd.

We had the principles of beginning class for a reason. It said any constant over any power of x as x approaches any infinity is going where, folks? Where? Zero. So where's this going?

Zero. Where's this going? Zero.

Where's this going? Okay. Come on, come on, everyone gotta play along.

Where's this going? Zero. Zero, zero, 15, zero. So this, after you stop running the limit, would be zero minus zero over 15 minus zero. That's where these things are going.

It's a constant over x as x approaches infinity. Negative infinity, but still an infinity. Zero, zero, zero, zero.

How much is that gonna be? Zero. That's zero.

That says a horizontal asymptote. That's what this is, by the way, a horizontal asymptote. as you're going to the left. Are you following me on that? Now, if you did this, if you reciprocated this, and you divide everything by x squared, do you see how you still have an x on the numerator?

That would be saying you're going to positive or negative infinity, depending on what x is approaching. Let's do one more. And then next time we'll start building on this concept. Two more, two more.

We've got three minutes now. Again, some of you who kind of understand this concept right now should probably be able to tell me where this is going to approach, or at least have a guess what it's going to approach. Positive and negative. Oh, okay.

Positive and negative. Okay. Ooh.

Well, you might want to do the work to make sure, but do you see it's not going to go to a constant? Do you see it's not going to go to zero? What are we going to divide by? X cubed, X squared, or X? The largest power in the denominator.

So show your work. Don't get lazy on that. Don't take a guess because some of you are guessing right now. Don't take a guess. Show your work.

Over x, over x, over x, all over. Now, divide everything by the largest power in the denominator, not the numerator. That'll give you undefined spots. Do you see that?

If I divide it by x cubed, I get this whole bunch of undefined stuff. Can't have that. Can't do that.

Are you alright with that one so far? Simplify them. You're going to get 7x squared minus 2x plus 1 over x. All over. 9 over x minus 2. You following on that?

Now let's kind of think carefully about what happens here. What happens? Where does that go? That goes to positive infinity. Does this stuff matter?

No. That goes to positive infinity. Agreed, it's x squared.

This is zero, but that is a negative two. Where's that going? Negative infinity. In fact, if you really wanted me to show it to you this way, I could.

Couldn't you factor out a negative and get 2 minus 9 over x? Move the negative up top. Negative infinity.

Bam. Got it. Did you see it?

Did I lose you? You understand I went to positive infinity? Little unorthodox here, but you see this goes to zero. Negative 2 doesn't change.

Positive divided by negative is a? Infinity. Infinity divided by 2? Still infinity. Okay.

Now mathematically, oh, not that. It doesn't do that. Math.

Mathematically, the way that you could show this is the way I showed you. If you factored out the negative, it's negative 2 minus 9 over x. Right?

If you take that negative and move it to the top of your fraction, which is legal to do, it goes there. That becomes 2 minus 9 over x. That says this goes to infinity times a negative, that's negative infinity.

Divided by a positive now, that's negative infinity. That's what that means as well. Can we show it the way it's on the left and on its way?

Yeah. Thank you. Yeah. By the way, did anything with the highest powers of x have anything to do with our problem here? This one and that one is really what caused this to be the way it was.

One last thing I'll leave you with, just this idea. Because I want you to look at some of your homework tonight, but I don't want you to be completely afraid of it. Just a little frame. That looks pretty nasty, but tell me something that I can do with exponents. What can you do?

Divide by exponents. Not yet. You can move them outside.

I can move them outside. Is a root a type of exponent? Then look.

You can take a cube root out of the limit. Huh? Now ignore the cube root for a second, just for a second though.

Can you do this limit? Yeah. Explain to me how you would do this limit. What would you divide by here?

X squared. X, which one? X squared.

X squared. Hey, where's this limit going to go from the, on the inside? Two thirds.

Do you see the two thirds? Two thirds. Two thirds? And then take a cube root of that. So our answer here, after you show the work, would be a cube root of two thirds.

The work you show is this. You show a cube root 2x squared over x squared minus 3 over x squared all over. That's what you show, but that's the answer you get.

How many people understood today? I feel okay with it. Right side people, are you guys doing this so far?

Yeah? Okay. Okay, so we're talking about how to do limits as x approaches infinity. Now the last example I gave you, I think I gave you that one, didn't I? I said that when you have a cube root, it is a power.

That means you can pull it outside of our limit and say, well let's take the cube root of our limit itself. Now when you do that, that's kind of nice, right? Because we know how to take these limits. What we normally do is divide everything by the largest power in the denominator. That's x squared.

You would get 2. That's going to be 0. 3. That's going to be 0. You're going to get 2 thirds. You follow? Don't forget about the cube root, but your answer is 2. Two-thirds with a cube root around it.

I think that's what I gave you last time. Were you okay with that one? Yeah.

Now, I showed you that one again because I want you to consider the problem that's just below it. Can I do the same thing here? For instance, can I pull out... the square root around the whole limit? No.

No, because that square root's not around this whole function, it's just on the numerator, so. Oh my gosh, well what do we do? What do we do? Well, let's try to stick with the normal operation of this.

When you take an x to infinity, we can't just go, well, that's infinity squared. Okay, that's still infinity. Plus 2 infinity, square root of infinity over infinity. One. We can't do that.

But what we did say was maybe we divide by the largest power of x in the denominator. So we're going to look down there. What's the largest power of x in the denominator?

x to the first power. So let's divide the numerator and the denominator by x. Are you okay with that so far?

Well now here's the issue, and some of you might see this issue already. Can I just take this x inside that square root? I can't unless I have a square root.

Remember, you can only combine things if you have the same exact type of root. Follow? Because it's an exponent.

You can only put things together if you have the same type of exponent. Well, that's a problem. Does anyone know how to change x into something that I can take into that square root? Let's think about it for a second.

Let's think about that x. What do you think? But if I do x squared, I have to change those to x squared. Right?

Is that a problem? That would be zero and zero. That would be undefined. So I can't just arbitrarily change x squared, but you're on the right track.

2 over 2 is still 1. x over y, I did 1 over x, I did that. So x squared down here, we can't have an x squared, I need this to still be a value of x. Make it a value of x squared, it's a great idea. x squared times x squared is x to the 4th, that's too much. Say what now?

Root of x times root of x is still x. You're close. Oh, you're so close. You're so close. Put that together.

You want something that equals x, true? How do you make this equal x? Do that, and they are equal. Is that equal? Okay.

No, no, but almost. That's our idea, though. I'm going to fix this in a second. So here's the plan.

Why no? that I need this. Do you see why I need that?

That is still, well, pretty much x, almost x. And that's going to allow me to take that into that square root. Do you follow me?

That's important. But the question is, is x equal to the square root of x squared? And the answer is no.

Not really. Because if you think about it, why don't you take like a negative number, negative 5. Is negative 5 equal to the square root of negative 5 squared? No, it's not. Because as soon as you square it, it becomes positive, right? So this is not necessarily true.

What is true is this. That's true. So if we have to have this, there's something that's wrong in our problem right now.

This is right. What is wrong? Do you see it?

You don't see it. Let me go through the process one more time. You knew maybe if I draw in purple, it'll make more sense.

You knew if you, you had to do that, right? But you knew that this wouldn't work. So to try to make it equivalent, you did this. That would go into the square root. However, this fails now because this is not equal.

That's not equal. Notice you're trying to divide everything by the same thing, right, the same thing. So if you have divided by the same thing, these are not the same thing. This. Is the same thing.

That's true. This is true. Yes, this is not true. Not necessarily true. So are the absent value...

This is almost true. Is this true? Well, no, almost.

This is true. You ready to keep going? Did you understand that idea right now? You sure?

Okay, so when we take the square root of x squared, we can't just have x, you have to have the absolute value of x. And I'll show you why it's important in maybe 10 minutes. For right now, let's continue working on this problem.

When you have a square root over a square root, that says you can take a square root of the whole entire fraction. That's x squared over x squared plus 2 over x squared. Are you okay on where the x squared over x squared and 2 over x squared are coming from?

That's the sole reason why you did that, right? Just to make that in there. Over 3x over the absolute value of x minus 6 over the absolute value of x.

Well, that's a limit. X goes to positive infinity. You're going to have the square root of 1 plus 2 over x squared.

All of them. Oh, okay. Now we've got to deal with that. I need your minds to think back to what it means to be an absolute value.

Absolute value is actually a piecewise function. The piecewise function says, if x is positive, you leave it alone. If x is negative, you change the sign. Does that make sense to you? If x is positive, positive, you leave it alone.

If x is negative, you change the sign. And that was the whole idea. That's the piecewise definition of absolute value. Do you remember that? I think we did in here, actually.

Well, This is how you apply that. We are going to have the 3x over minus 6 over. Now you get to change your absolute value of x depending on where you are going, whether you go into positive infinity or negative infinity. infinity.

Where are we going? Positive numbers or negative numbers? Positive. We're going to positive numbers say that the absolute value of x equals x. Positive numbers say that the absolute value of x equals x.

Are you with me on this? The absolute value of x. Now you don't need the absolute value.

You know that since I'm going to positive numbers that can become the x. It might seem trivial like well if it's just going to change to x anyway why do we need the absolute value? Trust me, you need them. I'll show you why in a couple minutes, okay?

Now I'm like, five minutes. Raise your hand if you feel okay with that so far. You see, now that we have that, those x's do match up. Those x's are gone.

You get a limit to positive infinity. One plus two over x squared, all over three. Minus six over x. Tell me some nice things that happen since x is going to positive infinity.

Tell me what happens. Where does this go? Not zero. This, the one. The one goes to one.

It's a constant. Do constants go anywhere? No.

Where does this go? Where does the fraction go? Have we lost it all? Oh gee, you're scaring me, people. You're scaring me here.

Limits of constants are constants. Limits of this is not undefined. Where's x going? x is going to infinity.

Where's this fraction going? Zero. Zero. A constant over infinity is zero. Divide a constant by infinity.

How much are you going to get? Zero. Zero.

Think about that. Come on. Where's three going?

Three. OK. Where's this going? Zero. Don't forget what we did last time.

This says you're going to do. One plus zero over three minus zero. Tell me my final answer, please. What is that? Are you okay with getting the zero and the zero?

I think we talked about that last time. A constant over any power of x as x approaches infinity is going to be zero. Because this number is getting huge, and this number is getting huge, and those numbers are staying the same. That means you're going to have a number over huge.

A number over huge is zero. Not the square root of 1, okay, 1 third. Show of hands, how many of you feel okay with that? Now, I want you to consider this.

Why this is important. Think about it. You can change your problem if you want to. I'm not going to do the whole example over again.

I'm not going to give you two examples. I'm going to show you with this. Let's say that that... It is now negative. That means that's negative.

Is this still possible? Absolutely. Is this still true? Absolutely.

Is that still okay? Yes. At this point, something different would happen.

At this point. From here to here. You see, if we're going to negative infinity, that means that for our absolute value of x, where are we going? We're not going here anymore.

We're not dealing with positive. numbers we're dealing with negative numbers. That means instead of absolute value of x equaling x, absolute value of x would equal negative x. That means that this and that would be there.

Do you see the change? This would be okay. Square, right?

Square's going to be positive no matter what. That's where our definition came from. But this would be different. What's 3x over negative x? Minus 6 over negative, well that's a plus.

That's not really an effect. that's being plus zero, but this definitely affects us, doesn't it? That's negative three plus zero.

That's negative 1 3rd, not positive 1 3rd. What this says is, this is interesting, the first time we've had this, because if you've been thinking about it, you might have said, I said, well, why do we even take positive infinity and negative infinity? Why don't we just do one infinity and consider it a horizontal asymptote the whole way? Well, it doesn't work.

Look at that. Our first example was positive infinity, right? That's that way. That was at a horizontal, and remember what these are? A horizontal asymptote?

That's a horizontal asymptote at 1 3rd. If you go to the left, negative infinity, is it at 1 3rd? No, it's at negative 1 3rd. So our horizontal asymptotes can be different as you go to positive infinity and negative infinity. That proves it.

That's different right there. How many people understood that? Cool.

This really is a real thing. Hopefully that made sense to you. We're going to do one more example, maybe two, just to start it off. Let's try that. I'll show you something kind of cool, interesting that you can do.

So we give it a try? Looks awesome to me. I don't know about you. Looks awesome to me.

Let's try. Ready? What's infinity to the fourth power?

Infinity. OK, cool. Plus 2?

And the square root of that? And then minus, so infinity, how much is infinity squared? 0. Done. Love that. Woo, easy problems.

Yeah? No. Of course, it may be coincidentally, but I really doubt it.

We can't do that. You can't just plug in infinities. Whenever you get a case where you're like infinity minus infinity, or infinity over infinity, something like that. that you know something bad's happening. You're not doing something right.

So you can't ever do that. You get infinity over infinity. Please, for heaven's sakes, don't put one.

Because that would have been one, right? We can't do that. What we do have to do is manipulate the problem somehow. Now in this case, you have a root. You have a root minus something.

What's one thing we like to do with roots minus something when we can't work on them any other way? We usually take... to try to rationalize it, all right?

Now this doesn't have a denominator, so make one. Put it over what? Put it over what? Get yourself a denominator. Then you're going to try to rationalize.

Conjugate. Very good. Conjugate.

You want to be fancy about it. Is it plus 2 or minus 2 there, by the way? Plus 2. Good, so this sign doesn't change. This sign does change.

And it's over exactly the same thing. Are you okay with that so far? Yes or no? Yes.

Well, the denominator has nothing to distribute, which is really nice. We don't have to do any work up there. The numerator, just distribute it carefully. You know with conjugates, your middle term should be bye-bye, right?

Your middle term should be gone. Also, the conjugate works this way because the square root times the square root gives you the radicand that's inside. That's nice.

So our limit to positive infinity becomes something over the square root of x to the fourth plus two plus x squared. The numerator becomes, oh, what's the numerator become? Distribute, right? Do you see that on your distribution? Because you should have those.

X to the fourth plus two. We're going to have plus and minus, the same exact thing. And then we're going to get, what's the last thing? You see the minus x to the fourth of 2? Don't forget about the exponent.

If you make the exponent incorrect, it's not going to come out right. So you really need to be careful about your distribution. Don't forget to distribute that last part. You got me?

What happens here that's nice? Yeah, oh that's great. I love that.

2 square root of... I'm going to go ahead and do that. x to the fourth plus two plus x squared. So far so good?

Okay. Can we do that legally? Can we just say two over a fifth? Well, I don't know. Especially if this is a, especially if you did this.

Especially if you did that, that would be a big problem. Do you see that? That would be an issue.

So what we're going to do is make sure about it. You're going to start dividing by the largest power in the denominator. Again, just like you normally would. So what's the largest power, especially if I give you this one, okay?

What if that was 2x? That would be a big problem. This one, yeah, Scott's right.

You probably could look at this and go, that's 2. That's infinity. That's infinity. There's no way that that's not going to be infinity. You got me?

this is going to go to zero. He's right. But to check your work, especially if you have something like this, that might not go to infinity.

That definitely wouldn't go to infinity. Okay, that wouldn't. So to check your work to show that, what you're going to do is go, all right, well, let's divide everything by the largest power in the denominator.

Now, don't trick yourself here. The largest power in the denominator is not x to the fourth. It's not x to the fourth.

What is it? The way you can think about it. Cover up the rest of it.

of the largest term, what's the square root of x to the fourth? It's x squared. That's your largest power.

It's the square root of that because it's inside the square root. So divide everything by that. I'm making you do this because I want you to see one more time this setup. Notice we're divided by x squared. If I divide this by x squared, is that going to work here?

Is this okay? No. Can I take that inside of that square root? Okay, so think back to what we did.

This is fine, right? This is fine. That's great. That's what we like.

What does this do? What now? Oh, okay.

We got stuttered again. Uh-oh. Square root of x to the fourth would be great.

Did you say that already? I was thinking, my mind was wandering. I was thinking about the Simpsons.

So, well since this is x squared and this is x squared, We need to make this equal to x squared, but it also has to have a square root around it. You follow? You can't just do this because that's x. So change the power. Is that true?

Now, some of you are going to ask me, well, wait a second Mr. Leonard, don't you have to have some absolute value here? Let me ask you this question. Is this always true?

That actually is always true, no matter what. No absolute value needed because you have all positive, all positive. It's always going to be the same. Does that make sense?

So we don't even need that for this case. So here we're going to have, I just wanted you to see that one time, 2 over x squared, all over, we're going to have the square root. If you think about it, this is going to be 1. Plus 2 over x to the fourth.

Did you see the 1 plus 2 over x to the fourth? Yes, no? We get the 1x to the fourth over x to the fourth plus the 2 over x to the fourth. That was one.

Show of hands, how many people feel okay getting down to that far? Oh, that's not many. Let's have some questions if you guys are not so okay on this.

Are you okay on rationalizing? Yeah? Are you okay on crossing those things out? That should be easy.

That's the fun part. Are you okay on dividing by x squared because that is the largest power in your denominator? Are you okay on doing the square root of x to the fourth because that's still x squared?

Then we take it inside, we have x to the fourth over x to the fourth, that's one. Two over x to the fourth, two over x to the fourth. x squared over x squared, that's one.

Where does this go, ladies and gentlemen, when we're taking it to infinity? Where does that go? Is it okay to have zero on the top of the fraction?

Where does this go? Where does this go? Where does this go?

Would you agree that this denominator goes to two? So this equals... 0 over 2 or 0. You're going to notice in this case, if I change that to a negative, nothing about this changes. I don't have any absolute value.

Nothing changes. That says the horizontal asymptote in both directions is at 0. That's what it's going to be. You following me on that one? Would you like to see one more case? Yeah.

Would that be all positive? ...experience? Or even experience?

Would that be the case? We just started with the evil one there. That was...I'm sorry, I'll go there.

That was evil. Oh, I don't know. It really depends on what you have.

I can't say a cover-all situation for every time. Should we change the problem just a little bit? Let's change the problem just a little bit, alright?

So what I'm going to do, instead of doing a whole completely new one, do you have any questions on this? We're going to see how it changes if I do this little bit to it. That. If that changes our problem at all. Let's see the things that change.

Let's go through our problem. I'll try to use a different colored pen so you see the differences. Can you still rationalize?

What's the only thing that changes about our rationalization? That's going to be an x squared. Are you all right with that so far?

What changes here, ladies and gentlemen, when I distribute? Just an x squared. You see it?

It would be kind of silly to do the whole example again, right? It's pretty much the same thing. That would be an x squared, and this would be an x squared.

You still okay? Well, what happens here then is this is an x squared, and that's an x squared. True?

Okay. So, okay, well, hang on a second then. Well, if that's the case, Are we still going to be dividing by x squared down here? Yeah, because that's still the largest power.

That's not. That actually, if you looked at it, would be x. The square root of x squared is x, so that wouldn't be the largest power.

We still divide by x squared, but now I have an x squared right there. That 2x squared, that's right there. This would look almost the same as if I have an x squared there.

What that means is I'll have a limit as x approaches positive infinity of 2 over, this is going to give you 1, plus 2 over x squared. You know I'll show all the work for you in case you're having trouble following me. X to the fourth over x to the fourth plus 2x squared over x to the fourth, x squared over x squared.

Are you guys satisfied with that one? Hopefully? Yeah. Okay, good.

This gives you 1, so we're still at 2. That gives you 1. This gives you 2 over x squared, and that gives you 1. We're now ready to take the limit. Ready to let the x go to infinity. What happens to our 2? Does it go to 0? It didn't last time because we had an over x.

Where does that go? Where does that go? Where does that go? What is it? We've got a 2 over the square root of 1 plus 0 plus 1 plus 0. 1 plus 0 plus 1. That's the square root of 1, that's 1. 1 plus 1 is 2. We get 2 over 2. Did it change?

So, a little bit. So that little change there said we're not at 0 anymore, we have a horizontal asymptote at now 1. So we're going at 1 when we get to infinity. How do you feel okay with what we've talked about so far? This is such good stuff.

Just kidding. I'm just blowing out. Allergies are getting to me. Okay, last one. Is it making sense so far?

Can you follow that? Yes. Shane, you okay?

Yeah. All right. Oh, that's nasty.

Still sure you're okay with that one? How about that one? A little tough? Conjugate. There's nothing to sing, just conjugate.

Here. There's nothing to distribute. On the top and the bottom, you're...

Just think about it for a second. No derivatives. Oh, good try though. When you don't know the answer, just derivative that. Good try.

This is one of those 5% Oh, man. Yeah, close. Huh?

Why? It's going to, you have a negative under the square root. Ah. So if I take really big positive numbers, subtract them from 7, are they going to be positive or negative?

And they're under a square root. Doesn't exist. That was the easiest problem of the day. Come on.

No, it doesn't exist. So think about your limits. I mean, you can think of them like that.

You have a 7 minus something huge. 7 minus something huge is something really negative. Square root of something negative is undefined.

The full limit does not exist. Does that make sense? Now, if I did this...

It's not harder. It's the same idea. You have 7 plus a really big number. The 7 plus a really big number squared over that is still a really big number.

It's infinity. Alright, so you can think of that. The only problems you really come up with is when you do things like infinity plus or minus infinity.

infinity, infinity over infinity, that sort of thing gets you. You can't do those. But if you know it's going to infinity or it's not going anywhere, that's fine.

That's okay. You can consider things like that. There's no denominator, nothing to divide by, you're good. Does that make sense? So negative infinity would still be?

This, no, this is actually. Positive. Yes. Do you feel okay with our limits?

Have I blown your minds yet today? So, when you have issues where you're subtracting infinities, rationalize them. If you have a square root, rationalize them. If you don't think through the limit what it means to go to infinity, consider the type of problems you have. You already know polynomials, any polynomial is going to go to positive or negative infinity, right?

We talked about that last time. Said, well, it follows really the leading coefficient, or leading term, sorry, the power of leading term. So, x cubed is going to go up, or if it's negative x cubed, it's going to go down, and things like that happen.

So, be prepared for those limits. Are there any questions before we continue? All right.

Transcript for:Understanding Limits and Asymptotes

Transcript for:
Understanding Limits and Asymptotes