 Hello and welcome to a screencast today about limits and more specifically, graphical limits. So what is a limit? I have a definition here that says a limit is the output value that the function approaches as the input value approaches a given value. So basically what that means is you're going to be given an x value and you're going to have to look at your graph and see where the outputs or the y values are approaching as you get really close to that x value. All right, so some notation here that I have is the limit as x approaches a of f of x equals l. So what that means is that the given value is your a, so that's the x value you're going to be looking at, and you're going to have to figure out the y value or the output, which is your l, that your y values are approaching. All right, graphical limits, so those are most interesting or most relevant to study when you've got a graph that's got something to it. So what I did was I googled calculus limits and graphs or some combination thereof and came up with this website here that you can take a look at on your own if you want more examples to practice from. Okay, so I found this really cool graph that's got some interesting features to it and some more stuff that we can talk about. So first let's start and take a look at what's happening in the function around 3. So x equals 3 is here. What's happening to our function at and around 3? Well, it looks like it's behaving pretty nicely, right? There's nothing really crazy going on. And so now let's take a look at the limit as x approaches 3. So what we want to do is we want to come into 3 from the left, so just get kind of close to it. So right in here, what does it look like my y values are approaching? It looks like they're approaching 1. Let's come in on the right side of 3, okay? So coming in here to the right side of 3, again, what are your y values approaching? It looks like they're approaching 1. So we can say that the limit as x approaches 3 of our function f of x equals 1. Okay, now let's take a look at 2. What's happening to our function around 2 or at 2? Oh, goodness, there is a big jump in the graph, okay? I've got a closed circle here in an open circle here, which we'll talk about in just a second. But I can definitely say that there is a jump at x equals 2. Okay, what can be said about the limit as x approaches 2? Well, let's do what we just did. So let's come in from the left 2. It looks like my y values are also approaching 2, but this is where the jump occurs. If we come into the right side of 2, it looks like my y values are now approaching 3. Okay, so what can we say about the limit as x approaches 2? It does not exist. I'm going to underline that not there, okay? So even though the function exists at 2, and that's our closed circle here, if you remember from your algebra days, if a closed circle means a function is included at that point, open circle means it is not, okay? So for this particular point, the function at 2 is 2, but the limit does not exist, because as you come in from the left and as you come in from the right, it's approaching 2 different values, and that's not the same thing. All right, the last one I want to look at here is what's happening to our function at 4. So at 4, we have a hole in the graph. So again, from your algebra days, that hole means our function does not exist. Okay, so here's the next question. When a function does not exist at 4, okay, in this particular one, that's the case. What can be said about the limit as x approaches 4 of our function? I'm giving you three options here. So I want you to pause the video, think about these three options, and see which one you think is going to be the best. Okay, the answer to this one is B. The limit does exist, because again, if you come into the left of 4, what are your y values approaching? They're approaching 2. As you come in from the right of 4, what are the y values approaching? They're approaching 2 still, the same number. So we can say that the limit as x approaches 4 of our function equals 2. Equals, there we go. Okay, so we've seen three cases here. We started off with the number 3, or the value 3, and we saw that the limit and the function both exist and happen to be the same thing. Then we looked at 2, where we saw that the function existed, but the limit did not because of a jump in our graph. And then we also looked at 4, where the function did not exist, but the limit did. So limits can be a little bit confusing in that sense, as they don't always behave like the function, but sometimes they do. So you just have to look at your graph, or as a table as we're gonna see later on. And just kinda see what's happening to the values, and then that will tell you what's going on. Thank you very much for watching.