Limits

This section expects students to have familiarity with the various ways we present multivariable functions including surface plots of \(z=f(x,y)\), contour plots, and tables. The preview activity is used to recall some important ideas from the limits of single variable calculus.

While many instructors skip over the idea of continuity, we present continuity of multivariable functions as a simple conceptual extension of continuity of single variable functions (limit exists at point, function exists at point, and limit and function agree at point). Additionally, we present polar coordinates as a tool to algebraically convert two variable limits into a one variable limit using a common geometric tool (another advantage of doing alternate coordinate systems with the precalculus materials).

In this section, we want to study limits of functions of several variables and will primarily look at limits of functions of two variables. In single variable calculus, we studied the idea of a limit, which turned out to be a critical concept which served as the critical third step in the to understanding the derivative and the definite integral. In this section, we will study how the concept of limit for functions of two variables is similar to what we encountered for functions of a single variable. We will use the notion of the limit of a function of several variables as the last step in understanding and defining important concepts related to differentiability later in this chapter.

We did not need to generalize our ideas related to limits when we looked at vector-valued functions of one variable in because we applied our one-variable limits componentwise. That is, the limit of the function \(\vr(t)=\langle x(t),y(t),z(t) \rangle\) is evaluated as three separate one-variable limits.

We begin by reviewing the idea behind a limit of a function of one variable. We say that a function \(f\) has a limit \(L\) as \(x\) approaches \(a\) provided that we can make the values \(f(x)\) as close to \(L\) as we like by taking \(x\) sufficiently close (but not equal) to \(a\). We denote this behavior by writing \[\begin{aligned}\end{aligned}\]. Remember that you may need to look at the outputs of the function \(f\) as you approach the input \(x=a\) from the right and left separately in order to ensure that the limit exists.

In Preview Activity, we used the notion of limit from single variable calculus and saw several different outcomes for limits in terms of tables and graphs. Specifically, when the limit \(\displaystyle{\lim_{x \rightarrow a} f(x)}\) exists, its value will be the value that the output of \(f\) approaches as the \(x\)-value inputs get closer to \(a\). We must look at whether the output of \(f\) approaches the same value as the input values approach \(x=a\) from different directions. Tables of values for the function can give evidence that a limit will either exist or not but will not have enough evidence to show a limit exists. Similarly, we can use graphs to visually represent the behavior of the output values as we approach a particular input from different directions.

Condensed — the full section is in Boelkins et al., Active Calculus Multivariable.

In this section, we will look at how the concept of limits on single variable functions can be expanded to work with functions of two variables. We will focus on two variables here because tools like tables and graphs readily make sense for a function of the form \(f(x,y)\) but become more difficult to use for functions of more than two variables. All of the concepts of limits and continuity can be expanded to functions of any number of variables but discussion of the details of this type of generalization will be deferred to a later mathematics course.

The biggest difference between limits of single-variable functions and multivariable functions is the number of ways we can approach a particular input. For a single-variable function, there are only two directions in which we can approach an input like \(x=a\), which we call approaching from the left and approaching from the right. We illustrate this in . When considering a limit of a single-variable function, we can evaluate and compare the output of a function in these two directions. If the output of the function does not agree as you approach from the left and right, then the limit does not exist, as we saw in .

In contrast to the single variable case, there are many different ways to approach a point \((a,b)\), when considering this point as the input for a function \(f(x,y)\). In Figure, we show seven different ways to approach the point \((a,b)\). When we look at how to get close to an input in two dimensions, we must think about more than just the straight-line directions. For example, we have to consider curved paths. In the next example, we will explore how we can think about the various representations of a multivariable function and what it would mean to examine the output of a multivariable function as the inputs get close to a particular value.

Our work in showed that the limit of \(f(x,y)=3-x-2y\) as \((x,y)\) approaches \((0,0)\) is 3. Symbolically, this looks like \[\begin{aligned}\end{aligned}\]

As we have seen in Activity, if \(f(x,y)\) has two different limits along two different paths as \((x,y)\) approaches \((a,b)\), then we can conclude that \(\displaystyle{\lim_{(x,y)\to(a,b)}f(x,y)}\) does not exist. This is similar to the one-variable example \(g(x)=x/|x|\) as shown in Figure; \(\displaystyle{\lim_{x \to 0}g(x)}\) does not exist because we see different limits as \(x\) approaches 0 from the left and the right.

Condensed — the full section is in Boelkins et al., Active Calculus Multivariable.

In , we saw how looking at all different types of paths that approach the input of interest is necessary to determine whether the limit exists. Additionally, shows how a careful set of algebra and inequalities can be used to prove a limit does exist. When working with limits in single variable calculus, you often either used properties like those listed above or had to use other tools related to measuring the rates of change in different functions.

One tool that will work well for functions with two independent input variables is polar coordinates. If we convert our problem (both the function and the limit) to polar coordinates with the point of interest at the origin, then we can evaluate our limit in terms of \(r\) going to zero. Remember from that polar coordinates allow us to separate the distance from the origin and the rotation around the origin. In other words, the \(r\)-coordinate measures how far we are from the origin (our limit point) and \(\theta\) will describe the direction with which we are approaching the origin (our limit point). In , there are several examples of converting points and equations into polar coordinates so will refer you there to review as necessary.

These examples may tempt you to conclude that polar coordinates will solve all problems related to finding limits of functions of two variables. However, that simplification misses the importance of how convenient it was to convert to polar coordinates in these functions. If we were examining a function that did not have a convenient conversion to polar coordinates or the algebra to simplify and evaluate the limit in polar coordinates was difficult, then polar coordinates will not be a great tool for evaluating limits. (Consider, for example, what would happen if you tried polar coordinates in .) Polar coordinates as a tool is presented here because it highlights how to geometrically separate position in two dimensions into a distance and a measure of directionality. These ideas recur in many places throughout this text and have already been used in our work on vectors. For instance, we often analyze the magnitude of vector measurements differently than we analyze direction aspects of vectors.

Condensed — the full section is in Boelkins et al., Active Calculus Multivariable.

Recall that a function \(f\) of a single variable \(x\) is said to be continuous at \(x=a\) provided that the following three conditions are satisfied:

1. \(f(a)\) exists,

2. \(\displaystyle{\lim_{x\to a}f(x)}\) exists, and

3. \(\displaystyle{\lim_{x\to a}f(x)=f(a)}\)

Using our understanding of limits of multivariable functions, we can define continuity in the same way.

For instance, we have seen that the function \(f\) defined by \(f(x,y) = 3x^2y^3 + 2xy^2 - 3x + 1\) is continuous at every point. And just as with single variable functions, continuity has certain properties that are based on the properties of limits.

Let \(f\) and \(g\) be functions of two variables that are continuous at the point \((a,b)\). Then

1. \(cf\) is continuous at \((a,b)\) for any scalar \(c\)

2. \(f+g\) is continuous at \((a,b)\)

3. \(f-g\) is continuous at \((a,b)\)

4. \(fg\) is continuous at \((a,b)\)

5. \(\frac{f}{g}\) is continuous at \((a,b)\) if \(g(a,b) \neq 0\)

Using these properties, we can apply results from single variable calculus to decide about continuity of multivariable functions. For example, the coordinate functions \(f\) and \(g\) defined by \(f(x,y) = x\) and \(g(x,y) = y\) are continuous at every point. We can then use properties of continuity listed to conclude that every polynomial function in \(x\) and \(y\) is continuous at every point. For example, \(g(x,y)=x^2\) and \(h(x,y)=y^3\) are continuous functions, so their product \(f(x,y) = x^2y^3\) is a continuous multivariable function.