All right. Folks. Welcome. Let's continue our study of limits. I'm going to start by introducing a function, one hopefully that we are familiar with. What if I say to you, consider the function the square root of X. Now, can you picture the graph in your head? Think of it. Hopefully, in your mind you have something that starts at the origin, goes off like that. Good. I want to talk about this function and I want to apply our new concept of limits to this function. But in particular, let X equal 0 or let X approach zero. Right away you see that there's a little bit of a problem, just a little bit, not a big one. We can easily overcome it, but that's what we're trying to get at. In the definition of a limit, you have to talk about approaching the point from both sides. You got to imagine I'm a little bug, antenna, legs. I'm walking down this curve to zero. I don't care what happens at zero. But then the definition of limit said from either side. Now I need to be a bug and I need to come at this point zero from the left. But there's a problem, like where do I draw the bug? A bug, it just falls down. Down it goes. Swat. We don't like to kill bugs here in math class. The problem is, it doesn't make sense to talk about the limit at zero for a function that doesn't have a two-sided thing, but there's nothing on the left or on the right. As you can tell by the definition of the section here, that gets us into what we call one-sided limits. You only need to consider for this one here, like from the right. Now it's a little tricky because as I'm coming in from the right, I actually move left. Just keep track of these things. What you talk about here, it's a quick little adjustment just to be technically correct, we're going to say the limit as X goes to zero. Now remember math people are super lazy. Writing from the right is too much work, too much time, so they put a little plus. When you see that little plus, it means from the right. Now I'm asking, instead of two bugs moving simultaneously to one spot, I say, where does the little bug want to go from the right as it moves left as I come in from the right. This one here wants to go right to zero. We have to be careful this is a domain issue. You can't plug in negative numbers. Just be careful with these little pluses for all the same reasons, you can also have, you can if you wanted from the left, just flip the graph or something like that, from the left. You would write that as the limit as X goes to zero with a little minus sign. Then whatever function you're talking about, and that would equal some limit or maybe it doesn't exist, I don't know. But for this particular example, the square root function, it wouldn't make sense. It just wouldn't make sense to say what's the limit as I approach zero, because that implies a two-sided limit, like as if coming in from the right and the left. It just doesn't make sense as a question. I would never ask that, and the same time it doesn't make any sense to say what is the limit from the left, what's the one-sided limit from the left for the square root graph? It also doesn't make sense. It has this intuition of just carrying about one side of this thing. The relationship between two-sided limits and one-sided limits is that, I'll put this down, one with little stars is important. For a limit as X goes to A, for specifics as the two-sided limit here without the little plus, without the minus. For this limit to exist, it must, super-duper double underlined here. It must be the same value L if you approach from the right or from the left. It can't matter. Basically, let's put this in symbols. The limit as X approaches A of a function is some number L if and only if, the one-sided limit as I approach A from the left is equal to L, and the one-sided limit, as I approach the value from the right, f of x is equal to l This is extremely important. If you have a function where you get different values as you approach from the right or left, then the overall limit, the two-sided limit, the thing without the little plus or minus on the value, it's that it doesn't exist. Let's see an example of that. Here's a function you may be familiar with or may be new, not sure. Let's do f of x equals the absolute value of x over x. If you haven't seen this one before, let's play around with it for a minute. What happens if I plug in one? It becomes the absolute value of 1 over 1, and that's just 1. If you plug in two, you get the absolute value of 2 over 2. Absolute values don't do anything to a positive number and they cancel. You'll quickly see, if you plug in any positive number, so for x positive, as I plug it in, I get the positive value. The absolute value doesn't do anything to the positive number. It's actually equal to just x over x and it will always cancel. So for x positive, I get one. Now, I can't plug in zero. There's a big goal, divide by 0 here. At zero, this thing is undefined, so open circle and then out it goes. What happens if I start plugging in negative numbers? If I plug in negative one, you get the absolute value of negative 1 over negative 1. Absolute value on the top kills the negative sign. You get 1 over minus 1, and you get minus 1. If I plug in negative two, something very similar is going to happen. You get 2 over negative 2, you get minus 2. The same thing is if you plug in a negative number, f of x becomes a negative x. The negative on the top cancels, you get positive divided by a negative that's always negative. So negative x over x, and that's just negative 1. This function looks like this. This is an interesting function. It's two lines. To the right of the y-axis, is one straight line and to the left, it goes at negative one. You can think of this like a piecewise function maybe. So it's one for x positive, negative one for x negative. At zero, it's undefined. You can't take 0 divided by 0. I've already yelled at you-all for that. Don't make me angry. This is the graph. Let's ask some questions about limits and plugging in. We evaluated this thing and that's great. Let's talk about limits here. What would the limit as I approach zero from the right of this function be? I'm approaching from the right. Some little bug, a little particle, big nose and I'm moving that way. I'm moving to the right. Where does the function want to go? Positive one. What is the one-sided limit of the function as I come in from the left? The minus sign means from the left. Now, I'm a little bugged down here and I'm moving this way. Where does the function want to go? it wants to go to negative one. Now, notice these are different values. If you put these two together, if I ask you what is the limit? As x goes to zero, that's the overall limit, the total limit, the two-sided limit. Because these two numbers are different, the one-sided limits are different, this limit of this function, absolute value of x over x, does not exist. The only way for it to exist is if both pieces are equal as you approach from the right and as you approach from the left. Super-important. Let's try one more. I'm going to do just a picture. No function. Let's say I have some like this. I don't know, open circle up there, close circle here, open circle here, and let's go down that way. This is a perfectly good piecewise function. We'll call it f of x, just to be different. Let's give this thing some numbers. Let's say here's 1, 2, picture not drawn to scale, 3, and we'll call this piece three. Why not? Let's purposely use some numbers that are the same. Here's a function. Let's ask a bunch of things. What is f of 3? Let's do some precalculus first. Let's evaluate a number. So where does the function actually go at three? At three, this function goes right where the hole is, and that's two. That's how you evaluate the function. Now, what's the limit? As x goes to 3, where does the function want to go? Not necessarily where does it go, where does the function want to go? Let's say, from the right of f of x? While you think of that, let's do the other one too. Where's the one-sided limit from the left? Then, what is the overall limit of this function? These are four separate questions that may or not be related to each other depending on how complicated the function is. Take a second to think about this. You tell me before I write it down. What's the limit as x goes to 3 from the right? Where am I going? If I'm a little bug, I'm going to move along, things are good. Where do I want to go? I want to go to 1. The function wants to go right to the value 1, the y-value. If I'm coming in from the left, I'm walking this way, where does the function want to go? Wants to go to y equals 3. These two numbers are different, 1 and 3. What does that mean about the overall limit, the total limit, what does that tell you? Since they are different, does not exist. If these two numbers are the same, let's say it's 1 and 1, or 2 and 2, or 3 and 3, then that is the value of the overall limit, something like that. Great. There's this another example. Now, you'll see questions for this section in two ways. One is that they'll give you graphs and ask you to compute these things, total limits, one-sided limits, evaluate the function, that sort of thing, get asked for domain, asked for range, all that stuff, but the other way to think about it is I can give you conditions and then you give me the graph, I'll give you the conditions on the right with limits and you give me the graph. These are little more open-ended, there's usually more than one answer to these, that's okay. Here we go. Here's the other type of question you will get. Sketch the graph of a function such that, now here's the conditions that I want, I want for limit as x goes to 0 from the left of the function, is minus 1, I want the limit as x goes to zero from the right of the function, is 2, I want the limit as x goes to 3 of the function to be zero, and then I want f of 0 to be 1. There's your conditions. I get more than one answer here. If you want, pause the video real quick and try this out. See if you can do it, and then come back and we'll check the answer. Ready? Here we go. Let's get some of the easy ones first, stuff we know how to do. F of 0 is 1, sure, right there, f of 0 is 1, done. The limit, now notice there's nothing here on the 3, so the two-sided limit, as I approach 3, one, two, three, has to be 0. That means as I come in from the right and I come in from the left, this limit is 0. Now, I could do an open circle here or I can do a closed circle, that doesn't matter, but somehow some way, this function has to be coming in to 0, or something like this. Open circle, closed circle here does not matter, but as I'm a little bug coming in from the right, coming from the two-sided limit, I have to go to 0. The limit as I approach 0 from the right is 2, that's up here. I already have a closed circle on 1, this one actually has to be an open circle at 2, and maybe we can connect the dots over here, you certainly don't have to. You could do piece-wise jump or discontinuity, but that's all good. As I approach 0 from the left, I'm heading down to minus 1, that has to be an open circle, there it is. Let's go this way. What you do after this tail here, it doesn't matter. But, now all your conditions are met. Anything you want that looks like this, hitting these conditions. You'll see questions both ways. Given the conditions, sketch to graph, or given the graph, computes the limits to do certain things. The big takeaway for this is that one-side limits are normal thing to talk about, especially with functions that don't have domain all reals, and you need both one-sided limits to be the same, if you're going to have a total limit, or if the two-sided limit is going to exist. Good job on this section. Try some more before moving on. All right, See you next time.
