 Hi, and welcome to a second screencast on how to compute the limit of a function using a graph. In the first screencast, we saw how to take a graph of a function y equals f of x and determine if that function has a limit as x approaches a particular number. The important idea was that the function f of x has a limit, let's say l, as x approaches a, if the outputs of the function approach l as x approaches a. The value of f of a may or may not exist, but as long as the function approaches a single value from the left and the right as x gets closer and closer to a, then the limit exists and equals l. In this screencast, we're going to start from functions given as algebraic formulas and then use a graphing tool to visualize them and then use the resulting graphs to determine whether the function has a limit at a particular point. Let's look at the function f of x equals x cubed minus 2x squared minus x plus 2 divided by x minus 2, and let's determine whether f of x has a limit as x approaches 2. First of all, notice that f of 2 doesn't exist, and we know that because if I substituted x equals 2 into the formula for f, I would end up dividing by 0. So we know that f of x does not exist at x equals 2. Now I have a question for you. What does this mean about the limit as x goes to 2 based just on the fact that f of 2 fails to exist, which is true? Can we conclude that the limit as x goes to 2 definitely does not exist, that the limit as x goes to 2 definitely does exist, or is there not enough information to tell? So pause the video and think about this multiple choice question, make your choice, and then unpause when you're ready to continue. So the answer here is C. Based on the information alone about the fact that f of 2 doesn't exist, we don't really know whether the limit exists as x approaches 2 or not. This is because the limit as x goes to 2 describes what the function is doing as x approaches 2. It's not about the behavior of the function at x equals 2. So in other words, we need to look a little deeper at this function and see what is happening to it as x approaches 2. So what we're going to do is use a graphing tool to turn this function into a graph, and then use the graph to determine if the limit exists like we saw in the first screencast. I'm going to be using GeoGeber as my graphing tool, which is a great piece of software that we will use a lot in this course, and you can download it for free at www.geogeber.org. You can also use a graphing calculator if you want, or a graphing calculator app for your phone or tablet, or an online graphing tool like desmos.com. Now to graph this function, I'm going to first open up GeoGeber and choose the Algebra and Graphics view. At the bottom in the input bar, I'm just going to type in the function f of x equals parenthesis x cubed minus 2x squared minus x plus 2 close parenthesis divide by parenthesis x minus 2 close parenthesis. Those parenthesis are very important, and then hit Enter. This is going to produce a basic graph of the function in the Graphics window on the right, and you can see the formula for the function in the Algebra view over here on the left. I'm going to click the arrow next to where it says Graphics, and then the little button that looks like a grid, and that will add a grid to this graph for easier reading. Now let's see if the limit exists of this function as x approaches 2. First of all, notice that if I type in f of 2 in the input bar and hit Enter, GeoGeber correctly tells me that f of 2 does not exist. So despite all appearances here, f of 2 really does not exist, and if we were drawing this graph by hand, we'd probably want to put an open circle right here at x equals 2. Now what matters for the limit calculation is what happens as x approaches 2, and it's very easy to see looking at this graph. It's quite clear that as x approaches 2, then the values of f of x are approaching 3, and it doesn't matter if you approach 2 from the left or from the right, it's the same value y equals 3 each way. So based on the graph, we can say that the function f of x has a limit as x approaches 2, and that this limit is 3. And we can say this even though f of 2 itself, at 2, the function does not exist. Now let's look at one more example, this time using what's called a piecewise function. A piecewise function is a function that has two or more different formulas that apply depending on the value of x. So here I have a piecewise function called g of x, which is defined to be x minus 2 if x is less than or equal to 1, and then it switches to x squared plus 1 if x is bigger than 1. Let's determine if g has a limit as x approaches 1, which is the point where the two formulas switch over. It's actually very easy to graph piecewise functions in GeoGeba much more so than on a graphing calculator, because we can graph each piece separately. Let's call this first piece a, and the second piece b. I'm going to go down to the input bar at the bottom of the GeoGeba screen and type in a equals function with a capital F square bracket x minus 2, which is the formula for the first piece, and then negative 3, 1. This is going to graph a function defined by the formula x minus 2, and the negative 3, 1 was going to restrict the domain so that x lies only between x equals negative 3 and x equals 1. Let's hit enter to see this piece of the graph. Now technically this piece of the graph should extend off to the left to infinity, but since in this example I'm only caring about the function near x equals 1, I'm just going to arbitrarily start this piece at x equals negative 3 because it's close to the edge of the window. Now let's graph the other piece by typing in b equals capital F function square bracket x squared plus 1, 1, 7, and hit enter. Now we see both pieces of the function g of x. What we care about is what is happening as x approaches 1, and it's very easy to see from this graph that as the values of x approach 1 from the left, we are getting values of the function g that approach 0. But as x approaches 1 from the right, we're getting values of g that seem to be approaching 2. Since g is definitely not approaching a single value as x approaches 1, we have to conclude that g does not have a limit as x approaches 1. Thanks for watching.