 Hello and welcome to this screencast on section 10.1, Limits. This screencast is going to cover limits of functions of two variables. Let's first review limits of a function of one variable. A function f has a limit l as x approaches the value a, provided that we can make the values of f of x as close to l as we'd like by taking x sufficiently close but not equal to the value a. We denote this behavior by writing the limit of f of x as x approaches a equals l. For example, consider the limit of the function f pictured to the left as x approaches 2. We say this limit equals 1 because we can make the values of f of x as close to 1 as we'd like by taking x sufficiently close but not equal to 2. Note that there are only two directions that we can approach the x value 2. We can approach it from the left, we can approach it from the right. This will be important to note later. The definition of a limit extends to functions of two or more variables. It's important to note that though we will focus on functions of two variables here, all the ideas we establish are valid for functions of any number of variables. For a function f of two variables, we say that f has a limit l as x, y approaches a, b, provided that we can make f as close to l as we'd like by taking x, y sufficiently close but not equal to the point a, b. We write the limit of f as x, y approaches a, b equals l. With functions of two or more variables, determining the behavior of limits gets more complicated. One reason for this is because with single variable limits, there are only two directions that x can approach the value a, from the left and from the right. With limits of functions of two variables, there are infinitely many directions that x, y can approach the point a, b. Since the domain of a two-variable function is two-dimensional, we can approach the point a, b from any of the surrounding directions. We will also see that the way we approach the point a, b matters. That approaching along a line is decidedly different from approaching along a curve. With all the different ways to approach the point a, b, showing a limit exists requires a more careful argument. On the other hand, as we will see in the activities for section 10.1, it does give us a way to show a limit does not exist. If we can find two different paths approaching the point a, b that result in two different limits, then we can conclude that the limit of f as x approaches a, b does not exist. This idea is similar to showing the limit of a single variable function does not exist by showing the left and right hand limits are not equal, like the limit as x approaches 0 for this function g. Many of the properties that are familiar from our study of single variable functions hold in precisely the same way for two variable functions. We can use these properties along with what we know from single variable calculus to show that certain limits exist. Feel free to pause the video here to look over these properties and take any notes. For example, consider the following two variable limit. Using properties two through five from the previous slide, we can break up this limit into these different limits that we have here. Then using what we know about single variable limits and property one from the previous slide, we can evaluate each of these limits to get here. And after a bit of algebra, we arrive at a final value of negative 17 over 6.