Hi everyone and welcome to our lecture on limits. We finally get to start talking about the heart of calculus, we get to do some calculus in this section. As I write the definition, keep in mind that limits are going to be the heart of what we study in this course, and in future calculus courses to follow. It is going to be the tool that allows us to overcome the division by zero problem, where algebra fails. This is the next level, this is the next tool that we need as mathematicians to go on and do calculus. We're going to write the limit as x approaches a. That's how you read this thing. Limit as x approaches a, of f of x equals L. L will be some number, a some number too. If we can make f of x as close to L as we like by taking x to be as close to a on either side, but not equal to a. There it is. It's a little disappointing. This is the thing that's going to make life great. This is the thing that's going allow us to get past all these walls, solve the tangent problem, solve the velocity problem. It is, but like any definition, is a little wordy and take some examples to sink in. Let's jump right to it. Let's do a specific example. How about f of x equals x squared? Let's put some numbers to a specific problem. As always, the rule, you have to pick this one first whenever you introduce a new topic. There's a good old parabola, and let's actually label this thing with some numbers. Here's 1, and here's 2, 4 is up here somewhere. Sounds good. This is a function. In the prior classes, we asked you to evaluate this function. We said, what's f of 2? That we said was 4. You learned how to do this and you got used to this notation with parenthesis, and realized this is not f times 2. There was a day when that was weird, and hopefully we're all past that. But now I'm going to ask you a different question. Instead of focusing exactly what happens at 2, I actually don't care anymore. In this case here, a is equal to 2. We pick a point and we just study it. I don't care what happens at the evaluation. This is a completely different way of thinking than the pre-calculus notion of plugging in, a little weird. We say don't you want to plug in? Not with limits, limits asked something else. They say, what if you're coming in, maybe you're a little bug or something, walking in the parabola from both directions. Imagine you're walking up, you're walking down. What limits ask is, where does the function want to go? As you approach 2, what does the function want to do, versus what is it actually doing? This notion is more powerful than you might realize at first. In this example here, the function wants to go right to 4. In this particular example, this easy example, this parabola, there's no difference between plugging in and taking a limit, but we'll see in a second that that could actually be different. Here, this is not anything new with these functions. It'll turn out if it's continuous, you won't have any problems. But let's do another example. Let me change this up a little bit. Let's do a hole in the graph or something , something like this. Here's a function, and we'll say we're 2 again. Here's 1, here's 2, and here's 3. Now, you the human can define the function. Imagine this is a piecewise function, where you just override the function. You're allowed to do that, you create whatever you want. If I asked you now to evaluate this function at 2. Here, we're just starting in the point a is 2. I said to you, what is the function doing at 2? You'll say, there's an open circle here. Maybe it's a piecewise function, we evaluate that, that's 3. The function is 3, and you're absolutely correct, that's fine. But now here's the different way of thinking. What does the function want to do? If I'm a little bug coming in this way, you have to look at it from both sides, that's the whole point of either side. Where does the function want to go? Another way to ask that is, what is the limit of the function as x approaches 2 of f of x? In this case, the function wants to go to two. Now you can see there's a difference between plugging in, evaluating the function, and taking a limit. This is the beauty of this thing. In fact, if you think about it, you don't even need the function to be defined. This is going to be the crazy part. If I remove this point entirely, I can't evaluate the function anymore. Just say there was a open circle, the function's not defined at two. Then asking what the value of the function is at two doesn't even make sense. However, I can still ask for the limit. You don't even need the function to be defined at the point to take a limit. That is going to be the key notion because what our goal is, if we want to overcome division by zero, the function won't be defined when I plug in zero. I can't just say what is f of 0 dividing by 0, but I certainly can say, what does the function want to do in terms of the graph? Keep that in mind. Let's do some more examples. Let's keep playing around with this. Get used to this thing. Here's another one. Let's do an example. We can use a graph. Let's look at the function f of x equals sine of x over x. Sine, everything's good, but it's a fraction again. So I'm going to focus on this division by zero at first. If I asked you what f of 0 is, you'd run into a problem. You'd run into a problem immediately because there's a division by zero over here. You can try to plug in and you're going to get sine of 0 divided by 0. I'm putting this in quotes so you realize that this is terrible and you can't do that. The rookie mistake, of course, I guess the smart rookie, the person who knows their trig, would say sine of 0? I know that, that's 0. Here it is 0 over 0 again, and I want to say this a million times, 0 over 0 is undefined. Do not tell me it is one, do not tell me it is zero, don't tell me it's Pi. This is undefined. Remember it's like a word that doesn't exist in the dictionary. It means nothing. This is, I want to put it in red here, not equal to zero, not equal to one, it's nothing, it's garbage. Sad. That's your official noise you make, this angry face, just mad. I think I'm angry because I've had like a million students in my life tell me this is zero and it's one, and I just get mad. Don't be that person, 0 over 0 is undefined. What does that mean? It means go try something else, it doesn't quite work. If we look at the graph of this thing and you can pull it up on some graphing website or your calculator or whatever. It does something like this at zero. It's undefined at zero, but the graph goes, there's a circle right there at zero, and it does its little arching thing. The height of this thing is one. F of 0, we'd say is undefined. You've run into division by zero, and there's where algebra stops. I can't divide by zero, life is over, done, close the book, go home, we're done. But then calculus comes along and it says, okay, fine. The function may not be defined at zero, but it looks like it wants to go somewhere. What is the limit as x approaches 0 of this function? What does the function want to do, even though it's unable to do so explicitly? Where does it want to go? This is the beauty of limits. I don't need the function to be defined at zero for me to answer this question. If you were a little bug, here's the bug, little head, antennas, and you're walking this way, and you're another little bug, really have two body parts or three body parts. Let's go two and a head, and you're walking that way. Where does this bug land? Where does this particle land for your physics folks out there who don't like bugs? The function wants to go to one. Looks like I can't talk about things when I divide by zero, but by looking at limits and by approaching a point from the right and the left and seeing where the function wants to go, I'm allowed to move on and keep studying this function near zero. Still can't plug in, but that's video limits is that you don't have to. We talk about the limit of this function. You can see that from the graph, you can play around with it, we can do a bunch of things. But our goal eventually is to develop and enhance our techniques to find limits using algebra and make algebra stronger. Right now we're guesstimating based on the graph, but it's a good place to gain some intuition. Another way to do this is you can also find limits using tables or calculator or other Excel properties , like numerical things. Let's do another one. If I said, take the function f of x to be the square root of, let's make it t so you don't get used to x, t squared plus 9 minus 3 all over t squared. This is a perfectly good function, everywhere you can evaluate this. But I, of course, I'm interested in, and this is where we're first going to start, what happens when t goes to 0? What happens is around 0?Hopefully you'll agree, if you plug in 0, the numerator becomes the square root of 9 minus 3, that's 3 minus 3, that's 0 again. And the denominator is 0 over 0, that's undefined. I know you're never going to tell me that that's 1 or something like that. That'll just make me angry, give me more gray hair. I don't know the graph of this one. I could go look it up and do that sort of thing, but let's try another way. What you can do is you can make a table and you can see what's happening, T-A-B-L-E. There it is, tough word to spell. Let's take some t values. Let's plug in the function and I worked these out before, but you can check these with your calculator. Because everything is squared, if you put plus or minus in, it's not going to change anything. What I'd like to do is get closer and closer to 0. Here's 1 and here's negative 1. Let's plug in and see what's going on there, at plus or minus 1. When you do that, you get 0.16228. Again, you go out as many decimals as you want, but you get some decimal, not surprising square roots, fine. Let's move in a little bit, so with a 0.1 is maybe like here. Let's move in a little bit. How about if we go to plus or minus 1/2? Put negative a 1/2 and positive 1.5 on the map, plus or minus a 1/2. And you can check if you plug in 0.5, plug in 1/2 to this thing, you get 0.16553. There's 0.16 is here and you can see it's approaching this 0.16 number. If you get even closer, so it's getting really close, going up a little bit. We can do plus or minus 1/10, and in that case it turns out, you can check with a calculator 0.1662, and you can get even closer, 0.05. Notice I'm getting closer to 0, but I'm never quite plugging in 0. Of course there are infinitely many numbers that are as close as you want to 0. It turns out if you get closer and closer and closer, there's 0.005, you get 0.16667. Eventually you start to give up and say, okay, this thing is approaching 1.666 and change. Your guess, again, you don't really know this, but we're guessing here, is that if I kept getting closer to 0, so as x gets closer to 0 from either the right or the left, doesn't matter, we seem to be approaching this 0.16 with a bar on it. We'll just keep going and going, going, and you can check that's 0.16. It seems to be approaching, you'd say the function f of x, wants to go to 0.16 with a bar, which is 1/6. This is our best guess, you can get this as accurate as you want to, as many decimals as you want. This is one way to use data to get a limit if you're not quite sure of what the graph is doing. Now, there are functions that we should talk about that still don't like when you play with limits with them. Let me do this as an example. How about the function f of x equals the sin of pi over x. If you get a chance, pause the video and go graph this thing in Desmos or something like that and play with it. I'm going to draw you a picture, and I see that division by x over here. Again, I'm interested in what does this function want to do as x goes to 0? Go graph this thing, play around with it. Because whatever I draw is not going to do it justice. Ignore that line. If you plugged it in 1, or something like that, or negative 1, and you get sin of pi. What's sin of pi? We all know that of course that's 0. You get 0 and 0. Let me change colors so we can see it. This function, it does something like this, and then it starts doing its little trig thing. It does something like this, starts doing its trig thing. But this is where I'm not going to Justice. Go look at your counting. If you want to break your calculator, you want to make it mad, plug this function in and zoom in around 0. Go to Desmos and go look at that thing. You'll see like this rectangle, this block, what's happening is the function is going bananas, as you get closer to 0. It goes up and down and up and down and up and down. In fact, it oscillates infinitely many times. It just doesn't like you dividing by zero here, it gets really angry and calculators have a hard time capturing this. It oscillates between one and negative 1 infinitely many times. The sine graph is bounded above and bounded below. This function goes crazy. It doesn't have a good behavior. Now how do you say? You say, okay, well I can plug in, so you say f of zero, which is undefined, you can't divide by zero, so here come limits, can I tell what the function wants to do? As I approach from the right, as I approach from the left, where does the function want to go? In this particular case, you can try to plug in the numbers you can see from the idea of this graph. It doesn't want to go anywhere. The closer using min, the more times that oscillates between one and negative 1, so just when you think it wants to go to one, boom, it switches direction and goes to negative 1. Just as you think it wants to go to negative 1, back to one it goes, it misbehaves. This function is so erratic near the origin that we would say this limit does not exist and we abbreviate that with DNE. You can only have one limit, there's only one number if it exists. If you're not making up your mind between one and negative 1, that's a big hard stop. That's a no. This function doesn't exist. Even though we have ideas of limits and where the function wants to go, then our functions out there, lots of them where the function just doesn't make up it's mind and we'll just say those limits do not exist. This is sort of the scary one. This is the one that we have it, it's out there, but then we quickly throw it in the rug so we don't scare you guys. Let's go back to the land of nice functions and just do two examples where function behave the way we want them to. I'm going to let you guys play with this for a second. Let's say what's the limit as x goes to zero of the constant function five? Let's just play around with this for a second. Think about this for a second. Let's get used to taking limits. This is a nice function. It's one of the nicest functions. How do you have a function that's nicer than a constant function? If I have five on the map, what's the line of the function f x equals 5? A straight line. As I approach zero, where does the function want to go? Do we all agree this is five? Okay. Doesn't it feel much better? We can breathe easier necessarily, so five. Let's do another one. That's not as scary when it bring everyone back down. We'll come back to the scary ones later, but let's get back to happier places. Go to your happy limit. How about this one, x squared minus 4? I think this one's easy enough that we can draw a graph. This is the parabola pulled down for y equals x squared minus 4, and I want to know what's going on at four, which is going to be out here somewhere, whatever. Where does the function want to go? Now I purposely drew this so that you don't see the graph, but hopefully you don't need to realize that this function, there isn't too much going on, it's a nice continuous function. It behaves the same as if you were just to evaluate it, so we actually extend the graph out here and drew it. This would become 4 squared minus 4 and that's 16 minus 4, which is of course 12. Now this just happens to be equal to f of 4 and that's okay. In the nice cases where the function works with the function you can evaluate, you can and up here by the way, this happens to be equal to f of 0 if your function is the constant function. This is our very simple approach, our naive approach to limits that's going to work only in the very beginning, as long as we stay sheltered from the complicated ones. But here's the first way to find limits, so you can check. You can find limits by plugging in. Now, plugging in when possible, let's just put it that way. Because as we saw before, especially if we're working around zero, it doesn't always work. You can't always plug in. If I'm looking at zero and there's a fraction and the dominant zero, I can't do it. But if I have something like x squared minus 4, life is good, simple, and we can do that. Some of the easier ones just to start off with, there'll be no difference between limits and evaluating, but that'll go away pretty quickly, so start off somewhere. Again, have a graphing calculator handy as you do it. Good job. See you next time.
