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 said 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 had 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's let x equals 0 or let x approach 0. 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've got to imagine I'm a little bug, antenna, legs, and I'm walking down this curve to 0. Again, I don't care what happens at 0. 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 0 from the left, but there's a problem; where do I draw the bug? Bug, it just falls down. Down it goes. 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 0 for a function that doesn't have a two-sided thing, that 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, 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 is a quick little adjustment. Just to be technically correct, we're going to say the limit as x goes to 0. 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 0. 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 if you wanted just flip the graph or something like that from the left. You would write that as the limit as x goes to 0 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 just wouldn't make sense to say what's the limit as I approach 0, because that implies a two-sided limit, as if coming in from the right and the left and it just doesn't make sense as a question, I would never ask that. At the same time, it doesn't make any sense to say 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 caring about one side of this thing. The relationship between two-sided limits and one-sided limits is that, and I'll put this down with a little star, it's important, as x goes to a for a specific side of the two-sided limit here without the little plus, without the minus, for this limit to exist, super-duper double underline 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, 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? Becomes the absolute value of 1/1, that's just 1. If you plug in two, you get the absolute value of 2/2. 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 just actually equal to just x over x and it will always cancel. For x positive, I get 1. Now, I can't plug in zero. There's a big goal, divide by 0 here. At zero, this thing is undefined. Open circle and then out it goes. What happens if I start plugging in negative numbers? If I plug in negative 1, 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 2, 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 negative x. The negative on the top cancels. You get positive divided by a negative that's always negative. 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, it's 1, straight line, and to the left it goes at negative 1. You think of this like a piecewise function maybe. It's 1 for x positive, negative 1 for x negative. At zero, it's undefined. You can't take 0 divided by 0. I've already yelled at y'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 0 from the right of this function be? I'm approaching from the right. I'm a little bug. Little particle, big nose, and I'm moving that way. I'm moving to the right. Where does the function want to go? Positive 1. What is the one-sided limit of the function as I come in from the left? The little minus sign means from the left. Now I'm a little bug down here and I'm moving this way. Where does the function want to go? Wants to go to negative 1. Now notice these are different values. If you put these two together, if I ask you what is the limit as x goes to 0, as 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 something like this. I don't know, open circle up there, close circle here, open circle here. 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. We'll call this piece 3. 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. Where does the function actually go at 3? At 3, this function goes right where the hole is and that's 2. That's how you evaluate the function. Now, what's the limit? As x goes to three, what does the function want to go? Not necessarily where does it go. What 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? and 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, think about this. You tell me before I write it down, what's the limit as x goes to three from the right? Where am I going if I'm a little bug, I get to move it along, things are good. Where do I want to go? I want to go to one. The function wants to go right to the value one, the y-value. If I'm coming in from the left, so I'm walking this way, where does the function want to go? Wants to go to y equals 3. These two numbers are different, one and three. 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 were the same, let's say it's one on one or two and two or three and three, 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, could ask for domain, ask 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 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, so that's okay. Here we go. Here's the other type of question. We'll get to sketch the graph of a function such that. Now here's the conditions that I want. I want the limit as x goes to zero from the left of the function is minus one. I want the limit as x goes to zero from the right of the function is two. I want the limit as x goes to three. I have a function to be zero. Then I want f of zero to be one. There's your conditions. This is all you'll 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 zero is one, sure right there. f of zero is one. Done. The limit, now notice there's nothing here on the three. The two-sided limit as I approach three has to be zero. That means as I come in from the right and I come in from the left, this limit is zero. 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 zero, 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, this is a two-sided limit. I have to go to zero. The limit as I approach zero from the right is two. That's up here. I already have a closed circle on one. This one actually has to be an open circle at two. Maybe we can connect the dots over here. You certainly don't have to. You could do piecewise jumper does continuity, but that's all good. As I approach zero from the left, I'm heading down to minus one. So that has to be an open circle. There it is. Let's go this way. What you do after this tail here, that doesn't matter. But now are your conditions are met, so anything you want that looks like this, hitting these conditions. You'll see questions both ways given the conditions gets to graph or given the graph of compute the limits to sad 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 or reals. You need both one-sided limits to be the same. If you're going to have a total limit, if the two sided limit is going to exist. Good job on this section. Try some more and then before moving on. See you next time.