 So in order to do calculus, we need to be able to find limits, and so let's start with multivariable limits, and let's keep in mind a limit must be the same no matter how you approach the limit. And as we saw, this can lead to problems when we have two or more variables. So let's think about this. The problem we ran into is that we let x, y go to zero any way it wanted to, and we are able to get different limits depending on how x, y went to zero. So maybe we can restrict how x, y is going to go to our point. And here's where we have to be a little bit careful. Once we do that, we are limiting the calculus we develop to certain types of functions. The good news is the types of functions we can apply our calculus to include pretty much everything you're familiar with. So far you've studied single variable calculus. That's the calculus of functions using one variable, like y equals x squared minus 4x plus seven. Wait a minute, aren't those two variables? So let's talk a little bit more about those variables. So remember a function takes inputs to produce a unique output. So the area of a circle is a function of its radius, and here we have one input, the radius, and so this is a one variable function. Or the area of a rectangle is a function of its length and its width, and so here we have two inputs, and so this is a two variable function. Or the votes of a politician are a function of their corporate sponsors, and here we have several inputs and we have a several variable function. So rather than letting x, y go to zero, zero at the same time, let's break it up. Let's find the limit as x goes to zero of the limit as y goes to zero of our expression. Now parentheses still mean the same thing they've always meant. Take care of the stuff inside first. So to find the limit as y goes to zero of this expression we'll let y go to zero. But what's happening to x? And at this point we can invoke a time on a tradition, procrastination. And in this case since we know nothing about what happens to x, leave it alone. So we want to find this limit, well x will leave as x, and y will let go to zero, and simplify. And so this limit as y goes to zero is one-third. And now we want to find the limit as x goes to zero of one-third, which is just going to be one-third. Well let's switch the order of the limit. So let's find the limit as y goes to zero of the limit as x goes to zero of the same mess. Well since the limit as x goes to zero of the limit of y goes to zero is one-third, then if I switch the limit around it's got to be the same thing, right? Well you're probably not taking multivariable calculus because you're bored and need something to do on a weekday night. Chances are somebody is going to do something with the problems that you solve. And here is the most important thing to keep in mind. If you don't find the flaws in your work, someone else will. So you really want to be careful about making statements like this without knowing that they are in fact true. Because if they're not true, somebody else will tell you that they are false. So let's check it out. So again, parentheses say do the stuff inside first. So we'll find the limit as x goes to zero of this expression. And again, since we know nothing about what happens to y, we'll leave it alone and get a limit of one-half. And then our limit of one-half will be one-half. And this problem illustrates a key point. Don't switch limit processes. What that means, a limit process is anything that involves a limit. These are things like, well, limits of course. Derivatives, don't forget that the derivative is defined in terms of a limit as is the integral and so are infinite series. And the important idea here is that if I have two limiting processes, for example, the limit of an integral, I can't generally switch them to be the integral of a limit. And if I do, I can't guarantee that they're going to be equal or in fact have anything to do with one another. And this is an important idea to keep in mind because you're about to have a bunch more limiting processes available to you.