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 sort of fails. This is the next level. This is the next tool that we need as mathematicians to go on and do calculus. So 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(x) = L. L will be some number, a some number, too. And if we can make f(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. All right, so there it is. It's sort of a little disappointing. This is the thing that's going to make life great. This is the thing that's going to allow us to get past all these walls these solve the tangent problems, solve velocity problem. It is, but like any definition, it's a little wordy and takes some examples to sink in. So let's jump right to it. Let's do a specific example how about f(x) = x squared? Let's put some numbers to a specific problem. All right, as always, the rule, you have to pick this one first whenever you're introducing a new topic. So there's a good old parabola, and let's actually label this thing with some numbers. So here's 1, and not great picture, 1, there's 1, and here's 2, so a 4 is up here somewhere. Okay, sounds good. So this is a function. In the prior classes, we asked you to evaluate this function. We said, hey, what's f(2)? And that, we said was 4. And you learned how to do this, and you got used to this notation with parentheses and realized this is not f times 2. And there was a day when that was weird, and hopefully we're all past that. But now I'm going to ask you different questions. So instead of focusing exactly what happens at 2, exactly what happens at 2, I actually don't care anymore. So in this case here a is equal to 2. We pick a point, and we just study it. So I don't care what happens at the evaluation. And this is a completely way of thinking than the pre-calculus notion of plugging in. It's little weird to say we don't want to plug in? Not with limits, limits ask 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. So imagine, [SOUND] you're walking up, [SOUND] you're walking down. What limits ask is where does the function want to go? Where does the function want to go? As you approach 2, what does the function want to do versus what is it actually doing? And this notion is more powerful than you might realize at first. In this example here, the function wants to go right to 4. So in this particular example, this easy example of this parabola, there's no difference between plugging in then taking a limit. But we'll see in a second that could actually be different. So here, this is not anything new with these functions. And 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. What if I did a function that looked like, I don't know, let's do a hole in the graph or something. So something like this. All right, so here's a function, and we'll say we're 2 again and at. So here's 1, here's 2, and here's three. Now, you, the human, can define the function. Imagine this is a p sub i function where you just sort of override the function. You're allowed to do that. You create whatever you want. So if I asked you now to evaluate this function at 2, right? So here we're just studying the point is a 2. And I said to you, hey, what is the function doing at 2? You say, there's an open circle here. Maybe it's a p sub i 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, and I'm a little bug coming in this way, where does the function, you have to look at it from both sides is 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(x). In this case, the function wants to go to 2. So now, you can see there's a difference between plugging in, evaluating the function, and taking a limit. And this is the beauty of this thing. So, in fact, if you think about it, you don't even need the function to be defined. This is going to be sort of a crazy point. If I remove this point entirely, I can't evaluate the function anymore. Just say there was an open circle, the function is not defined at 2, then asking what the value of the function is a 2 doesn't even make sense. However, I can still ask for the limit. So you don't even need the function to be defined at the point to take a limit. And that is going to be the key notion, because what our goal is to, if we want to overcome division by 0, the function won't be defined when I plug in 0. Or I can't just say, what is f(0) fill it up dividing by 0. But I certainly can say, what does the function want to do if, in terms of the graph, so keep that in mind. Let's do some more examples. Let's keep playing around with this schedule for this thing. Here's another one. Let's do an example. We can use a graph. Let's look at the function f(x) equals sine of x over x. All right, sine, everything is good, but it's a fraction again, right? So I'm going to bring up, I'm going to focus on this division by 0 at first. If I asked you what f(0) is, you'd run into a problem. You'd run into a problem immediately, because there's a division by 0 over here, right? You can try to plug in and you're going to get sine of 0 divided by 0. Putting this in quotes, so you realize it, 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 assign a 0. I know that, that's 0. And 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 1, do not tell me it is 0, don't tell me it's pie, this is undefined. Numbers like a word that doesn't exist in the dictionary, it means nothing. So this is, I want to put it in red here, not equal to 0, not equal to 1, not equal, it's nothing, it's garbage, sad. This is your official's noise, you make this angry face, just mad. I think I'm angry because I've had like a million students in my life, they tell me this is 0, and it's 1, and I just get mad. So don't be that person. 0 over 0 is undefined. What does that mean? It means we try something else, it doesn't quite work. So, if we look at the graph of this thing and you can pull it up on some grabbing website, or you calculate, or whatever. It does something like this at 0, it's undefined at 0. But the graph goes, is a circle right there at 0, and it does its little arching thing. And the height of this thing is 1. So f(0), we'd say is undefined. You run into division by 0, and there's where algebra sort of 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 says, okay, fine, the function may not be defined as 0, but it looks like it wants to go somewhere. What is the limit as the 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? And this is the beauty of limits, I don't need the function to be defined at 0 for me to answer this question, right? If you're a little bug, here is a bug, two little head antennas, and you're walking this way. And you're another little bug. Really have to 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. And the function wants to go to 1, right? So it looks like I can't talk about things when I divide by 0. 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 sort of move on and keep studying this function near 0. Still can't plug in, but this video limits is that you don't have to, so 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, define limits using algebra, and make algebra stronger. Right now we're sort of 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 sort of properties, is like numerical things. So let's do another one. So if I said, take the function, f(x) to be the square root of, let's make it t, just so you don't get used to x, t squared + 9- 3. All over t squared. Okay, this is a perfectly good function everywhere you can evaluate this, but I of course am interested in, and this is where we're first going to start, what happens when t goes to 0? Hopefully you agree if you plug in 0, the numerator becomes the square root of 9- 3. That's 3- 3. That's 0 again, and the denominator is 0. 0 over 0, that's undefined. I know you're never going to tell me that that's 1 or something like that. Just make me angry. Give me more gray hair. So 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 kind of see what's happening, T-A-B-L-E. There it is, tough word to spell. So 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. So what I'd like to do is get closer and closer to 0. So here's 1, and here's -1. Let's plug in and see what's going on there at + or- 1. When you do that, you get .16228 and you can 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 was that .1 is maybe like here. Okay, so let's move in a little bit. How about if we go to like + or- a half? So I put- a half and positive a half on the map, + or- a half. And you can check. If you plug in .5, plug in a half to this thing, you get .16553. Say okay, there's .16 is here and you can see it's like it's kind of approaching this .16 number. If you get even closer, so it's like getting really close going up a little bit, we can do + or- a tenth. And in that case, it turns out you can check with the calculator, .1662. And you can get even closer, .05. So notice like I'm getting closer to 0, but I'm never quite plugging in 0. And of course there are infinitely many numbers that are as close as you want to 0, and it turns out if you get closer and closer and closer, here's 5.0 or 5, you get .16667. And eventually you sort of give up and say okay, this thing is approaching like 1.666 and change so 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 .16 with the bar on it. We'll just keep going and going and you can check, that's one sixth. So it seems to be approaching, you'd say the function f(x) sort of wants to go to .16 with the bar which is one sixth. And you know this is our best guess. And you can get this as accurate as you want to as many decimals as you want. So this is one way to use sort of data to get a limit. If you're not quite sure what the graph is doing. Now there are functions that we should talk about that sort of still don't like when you play with limits with them, so let me do this as an example. How about the function f(x) = the sine of pi over x. If you get a chance, pause the video and go graph this thing in Desmos or something like that and just play with it. I'm going to draw a picture and I see the division by x over here. So again I'm interested in what does this function want to do as x goes to 0. So go graph this thing, play around with it, because whenever I draw is not going to do it justice. Okay, ignore that line. Okay, anyway. If you plug in 1 or something like that, or -1, then you get sine of pi. What's sine of pi? We all know that, of course, that's zero, so you get 0 and 0. Let me change colors so we can see it. So this function, it does sort of something like this. And then it starts doing its little trick thing. And it does something like this, starts doing this trick thing, but this is where I'm not going to do it justice. 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. It is, you'll see like 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 0. I think it's really angry, and calculators have a hard time capturing this. So it oscillates between 1, and -1 infinitely many times, the sine graph is bounded above, and bounded below. So this is function goes crazy, it doesn't have a good behavior. So now how do you say? So you say, okay, well I can plug in. So you say like F of 0, which is this is undefined, you can't divide by 0. So here come limits, can I tell what the function wants to do? As I approached in the writers, I approached on the left. Where does the function want to go? In this particular case, you can try to plug it in numbers, you can kind of see from the graph the idea of this graph. It doesn't want to go anywhere. The closer you Zoom in, the more times it oscillates between 1 and -1. So just when you think it wants to go to one boom, it switches direction goes to -1. Just as you think it wants to go to -1, back to 1 it goes, misbehave. This function so erratic near the origin, that we would say, this limit does not exist, and we abbreviate that with DNE. Okay, so you can only have one limit, there's only one number if it exists. If you're kind of not making up your mind between, 1 and -1, that's a big heart stop, that a no, okay? So this function doesn't exist, so even though we have ideas of limits, and where the function wants to go. There are functions out there, lots and lots of them, where the function just doesn't make up his mind, and will just say those limits do not exist, okay? So this is sort of the scary one, this is the one that we have, it is out there. But then quickly throw it into the rugs, we don't scare you guys. So let's go back to the land of nice functions, and just do two examples, where functions sort of behave the way we want them to. And I'm just going to do, I don't know what you guys put this for a second. So let's say, what's the limit as x goes to 0 of the constant function 5. What is fire out this for a second? So think about this for a second. Let's get used to taking limits 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 5 on the map, what's the line of the function F of x equals 5? Straight line, so as I approach 0, where does the function want to go? Do we all agree this is 5? Okay, it doesn't feel much better, we can sort of breathe easier. Nice sun, really, so 5, let's do another one. That's not as scary, when it bring everyone back down, will come back to the scary ones later. But let's get back to happier places, go to your happy, happy limit. How about this one, x squared minus 4? I think this ones easy enough that we can draw graph. This is the parabola pulled down for y equals x squared, minus 4 and I want to know what's going on at 4. 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're just to evaluate it. So we actually extend the graph out here and drew it, this will become 4 squared minus 4. And the notion is that 16 minus 4, which is of course 12. Now, this just happens to be equal to F of 4, and that's okay. So in the nice cases, where the function works for the function, you can evaluate, you can and up here. By the way, this is this happens to be equal to like F of 0. So you can check, you can find limits. Can find limits. By plugging in. Now, plugging in when possible, let's put it that way. Because as we saw before, especially if we're working around 0, it doesn't always work. You can't always plug in, so if I'm looking at 0, and there's a fraction of the number is 0 I can't do it. But if I had something like, I squared minus 4, life is good, simple and we can do that. So some of the easier ones just to start off with, there will be no difference between limits and evaluating. But that will go away pretty quickly. So start off somewhere and again, have a graphing calculator handy as you do it, all right. Good job, see you next time.