Motivating Questions
-
What techniques can we use to show that a function of two variables does not have a limit at a point \((a,b)\text{?}\)
-
What does it mean for a function \(f\) of two variables to be continuous at a point \((a,b)\text{?}\)
Section 11.2 Limits
Subsection 11.2.1 Introduction
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 Classic Calculus Approach 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 Chapter 10 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 \(a\text{.}\) We denote this behavior by writing
\begin{equation*}
\lim_{x\to a}f(x) = L\text{.}
\end{equation*}
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.
Preview Activity 11.2.1.
In this Preview Activity, we will consider several ideas related to limits of single-variable functions by working with tables, graphs, and the algebraic forms of these functions.
(a)
| \(x\) | \(-0.2\) | \(-0.1\) | \(-0.01\) | \(0.01\) | \(0.1\) | \(0.2\) |
| \(f(x)\) |
(b)
Write a sentence about what the table in part a suggests regarding \(\displaystyle\lim_{x\to 0}f(x)\text{.}\)
(c)
Write a couple of sentences to explain how your values in the table in part a and \(\displaystyle\lim_{x\to 0}f(x)\) are demonstrated in the graph below.
(d)
Let \(\displaystyle g(x) = \frac{x}{|x|}\text{.}\) Fill in the blanks of the table below with the appropriate outputs of \(g\) for values near \(x = 0\text{.}\) Note that \(g\) is not defined at \(x=0\text{.}\)
| \(x\) | \(-0.1\) | \(-0.01\) | \(-0.001\) | \(0.001\) | \(0.01\) | \(0.1\) |
| \(g(x)\) |
(e)
Write a few sentences about what the values in the table in part d suggests about \(\displaystyle\lim_{x \to 0}g(x)\text{.}\)
(f)
Write a couple of sentences to explain how your values in the table in part d and
\begin{equation*}
\lim_{x\to 0} g(x)
\end{equation*}
are demonstrated in the plot below.
(g)
(h)
Show that the limit as \(x\) goes to \(-1\) of \(h(x)\) exists. Write a sentence or two about why \(h(-1)\) does not exist but the limit as \(x\) goes to \(-1\) does exist.
(i)
Graph \(h(x)\) for a region around \(x=-1\) and explain how this graph relates to your answer to the previous two parts.
In Preview Activity 11.2.1, 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 \(x\)-values input get closer to \(a\text{.}\) We must look at whether the output of \(f\) approaches the same value 0as 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.
Subsection 11.2.2 Limits of Functions of Two Variables
In this section, we will look at how the concepts 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 \(x=a\text{,}\) which we call approaching from the left and approaching from the right. We illustrate this in Figure 11.2.1. 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 part 11.2.1.e.
A plot of two vectors on the number line approaching a point from the left and right
In contrast to the single variable case, there are many different ways to approach a point \((a,b)\text{,}\) when considering this point as the input for a function \(f(x,y)\text{.}\) In Figure 11.2.2, we show seven different ways to approach the point \((a,b)\text{.}\) 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.
The point \((a,b)\) is indicated in the first quadrant. Seven paths are illustrated approaching the point. Four approaches are paralell to the coordinate axes, approaching \((a,b)\) from the left, right, above, and below. One other path is a straight line, coming in diagonally from the upper left. The other two paths are curved. One is approximately parabolic and approaches from the upper right. There is also a path approaching from the lower left that bends several times as it approaches the point.
Example 11.2.3.
In this example, we will use tables and graphs to examine a function of two variables just as you did in Preview Activity 11.2.1 for functions of one variable. Here we consider the output of the function \(f(x,y)=3-x-2y\) approaches as the inputs get close to \((0,0)\text{.}\) Table 11.2.4 shows a spreadsheet of output values for \(f\) near the input \((0,0)\text{.}\) We have purposefully omitted the value of \(f(0,0)\) because we need to look at what is happening to \(f\) near \((0,0)\text{,}\) not what is happening at \((0,0)\text{.}\)
| \(x \backslash y\) | \(-1.0\) | \(-0.1\) | \(0.0\) | \(0.1\) | \(1.0\) |
| \(-1.0\) | \(6\) | \(4.2\) | \(4\) | \(3.8\) | \(2\) |
| \(-0.1\) | \(5.1\) | \(3.3\) | \(3.1\) | \(2.9\) | \(1.1\) |
| \(0.0\) | \(5\) | \(3.2\) | \(\) | \(2.8\) | \(1\) |
| \(0.1\) | \(4.9\) | \(3.1\) | \(2.9\) | \(2.7\) | \(0.9\) |
| \(1.0\) | \(4\) | \(2.2\) | \(2.0\) | \(1.8\) | \(0\) |
Table 11.2.4 shows that as the inputs \((x,y)\) get closer to \((0,0)\text{,}\) the output of \(f(x,y)\) gets closer to 3. We can look at some aspect of directionality using a table, like how the row corresponding to \(x=0\) shows values of \(f\) for inputs on the \(y\)-axis and the values on the diagonal (from top left to bottom right) shows values of \(f\) for inputs with equal \(x\)- and \(y\)-coordinates.
Visual representations like contour graphs and surface plots will give more information with respect to the different directions we can approach \((0,0)\text{.}\) Figure 11.2.5 shows a contour plot of \(f(x,y)=3-x-2y\) where you can use the slider to change the number of contours used in the plot. The input point \((0,0)\) is highlighted in red and you can see that regardless of how many contours are displayed, the output value of \(f\) seems to approach 3 near the red point. Just as with the table, you can get some idea on the directionality of how the outputs will approach 3, but because the contour plot does not give information about what is happening between contours, you are left to mentally fill in this information.
A plot of the surface \(z=f(x,y)=3-x-2y\) will give more information about what is happening in different directions as the inputs approach \((0,0)\text{.}\) The drawback to using surface plots is that they are difficult to create by hand. Even computer-generated plots of surfaces can be difficult to interpret and are prone to errors because of imprecise numerical calculations. In our case, the plot of \(z=f(x,y)=3-x-2y\) is a plane, which we have experience with from earlier in the course. We can see in Figure 11.2.6 that if we approach the input point \((0,0)\) in any direction or along any path, then the output of \(f\) (as shown by the \(z\)-coordinate on the plot) gets closer to 3.
If you were not convinced that the output of \(f\) approaches 3 as \((x,y)\) approaches \((0,0)\text{,}\) you could look at approaching the origin along a particular path. For instance, we could look to simplify our limit to look at approaching the origin along the path given by \(y=2x\text{.}\) This means that we don’t really need to examine our function’s output at all points around the origin. Instead, we consider only points of the form \((x,2x)\text{.}\) The limit expression becomes a one-dimensional problem in this case because along the path given by \(y=2x\text{,}\) \(\displaystyle\lim_{(x,y)\to (0,0)} f(x,y) \) becomes \(\displaystyle\lim_{(x,2x)\to (0,0)} f(x,2x)\text{,}\) which is the same as \(\displaystyle\lim_{x \to 0} 3-x-2(2x) \text{.}\) Note that the limit along this path is now of the type you evaluated in your single-variable calculus course. In fact, you can evaluate \(\displaystyle{\lim_{x \to 0} 3-x-2(2x)}\) to get \(3\text{.}\) It would be possible to use other paths such as \(x=\sin(y)\) or even a parameterized path of the form \(\vr(t)= \langle t \cos(t) , t \sin(t) \rangle\) to consider a two-variable limit. However, we can never test all possible paths that approach this limit point. Thus, while the idea of reducing a two-variable limit to a one-variable limit along a particular path can be helpful, we will not be able to exhaust all possible ways to approach our limit point and show a limit exists.
In all of our representations, we see evidence that if the input of our function, \((x,y)\text{,}\) approaches \((0,0)\text{,}\) then the output of our function goes to 3, regardless of the direction or way we approach \((0,0)\text{.}\) We generalize this idea with the following definition.
Definition 11.2.7.
Given a function \(f = f(x,y)\text{,}\) we say that \(f\) has limit \(L\) as \((x,y)\) approaches \((a,b)\) provided that we can make \(f(x,y)\) as close to \(L\) as we like by taking \((x,y)\) sufficiently close (but not equal) to \((a,b)\text{.}\) We write
\begin{equation*}
\lim_{(x,y)\to(a,b)} f(x,y) = L
\end{equation*}
Our work in Example 11.2.3 showed that the limit of \(f(x,y)=3-x-2y\) as \((x,y)\) approaches \((0,0)\) is 3. Symbolically, this looks like
\begin{equation*}
\lim_{(x,y)\rightarrow(0,0)} 3-x-2y = 3
\end{equation*}
To investigate the limit of a single variable function, \(\displaystyle{\lim_{x\to a}f(x)}\text{,}\) we often consider the behavior of \(f\) as \(x\) approaches \(a\) from the right and from the left. Similarly, we may investigate limits of two-variable functions, \(\displaystyle{\lim_{(x,y)\to(a,b)} f(x,y)}\) by considering the behavior of \(f\) as \((x,y)\) approaches \((a,b)\) along a particular path. Remember that the multivariable function situation is more complicated because there are infinitely many ways in which \((x,y)\) may approach \((a,b)\text{,}\) so we will not be able to check every possible path. In the next activity, we see how it is important to consider a variety of those paths in investigating whether or not a limit exists.
Activity 11.2.2.
Consider the function \(f\text{,}\) defined by
\begin{equation*}
f(x,y) = \frac{y}{\sqrt{x^2+y^2}}\text{.}
\end{equation*}
The graph of this function is shown below.
(a)
Is \(f\) defined at the point \((0,0)\text{?}\) What, if anything, does this say about whether \(f\) has a limit at the point \((0,0)\text{?}\)
(b)
| \(x\backslash y\) | \(-1.000\) | \(-0.100\) | \(0.000\) | \(0.100\) | \(1.000\) |
| \(-1.000\) | \(-0.707\) | — | \(0.000\) | — | \(0.707\) |
| \(-0.100\) | — | \(-0.707\) | \(0.000\) | \(0.707\) | — |
| \(0.000\) | \(-1.000\) | \(-1.000\) | — | \(1.000\) | \(1.000\) |
| \(0.100\) | — | \(-0.707\) | \(0.000\) | \(0.707\) | — |
| \(1.000\) | \(-0.707\) | — | \(0.000\) | — | \(0.707\) |
Based on these calculations, state whether you think \(f\) has a limit at \((0,0)\) and give an argument supporting your statement.
(c)
Now we formalize your conjecture from the previous part by considering what happens if we restrict our attention to different paths. First, we look at \(f\) for points in the domain along the \(x\)-axis. That is, we consider what happens when \(y = 0\text{.}\) Write out what the output of \(f\) will when \(y=0\text{.}\) In other words, what is \(f(x,0)\) equal to?
What is the behavior of \(f(x,0)\) as \(x \to 0\text{?}\) If we approach \((0,0)\) by moving along the \(x\)-axis, what value do we find as the limit?
(d)
What is the behavior of \(f\) along the line \(y=x\) when \(x \gt 0\text{.}\) That is, what is the value of \(f(x,x)\) when \(x>0\text{?}\) If we approach \((0,0)\) by moving along the line \(y=x\) in the first quadrant, what value do we find as the limit?
(e)
In general, if \(\displaystyle{\lim_{(x,y)\to(0,0)}f(x,y) = L}\text{,}\) then \(f(x,y)\) approaches \(L\) as \((x,y)\) approaches \((0,0)\text{,}\) regardless of the path we take in letting \((x,y) \to (0,0)\text{.}\) Based on the last two parts of this activity, can the limit
\begin{equation*}
\lim_{(x,y)\to(0,0)} f(x,y)
\end{equation*}
exist? Write a sentence or two to explain your reasoning.
(f)
Substitute \(y=mx\) into \(f(x,y)\) and simplify your result for \(f(x,mx)\text{.}\) Use your expression and limit rules to evaluate
\begin{equation*}
\lim_{x\rightarrow 0} f(x,mx)\text{.}
\end{equation*}
Use this limit to explain the the behavior of \(f(x,y)\) as \((x,y)\) approaches \((0,0)\) along lines of the form \(y=mx\) and how this observation reinforces your conclusion about the existence of \(\displaystyle{\lim_{(x,y)\to(0,0)}f(x,y)}\) from the previous part of this activity.
As we have seen in Activity 11.2.2, if \(f(x,y)\) has two different limits along two different paths as \((x,y)\) approaches \((a,b)\text{,}\) 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 11.2.8; \(\displaystyle{\lim_{x \to 0}g(x)}\) does not exist because we see different limits as \(x\) approaches 0 from the left and the right.
As a general rule, we have the following fact:
Limits along different paths.
If \(f(x,y)\) has two different limits as \((x,y)\) approaches \((a,b)\) along two different paths, then \(\displaystyle{\lim_{(x,y)\to(a,b)}f(x,y)}\) does not exist.
As the next activity shows, studying the limit of a two-variable function \(f\) by considering the behavior of \(f\) along various paths can require subtle insights.
Activity 11.2.3.
Define the function \(g\) by
\begin{equation*}
g(x,y) = \frac{x^2y}{x^4 + y^2}\text{.}
\end{equation*}
We will investigate the limit \(\displaystyle{\lim_{(x,y)\to(0,0)}g(x,y)}\text{.}\)
(a)
What is the behavior of \(g\) on the \(x\)-axis? That is, what is \(g(x,0)\) and what is the limit of \(g\) as \((x,y)\) approaches \((0,0)\) along the \(x\)-axis?
(b)
What is the behavior of \(g\) on the \(y\)-axis? That is, what is \(g(0,y)\) and what is the limit of \(g\) as \((x,y)\) approaches \((0,0)\) along the \(y\)-axis?
(c)
What is the behavior of \(g\) on the line \(y=mx\text{?}\) That is, what is \(g(x,mx)\) and what is the limit of \(g\) as \((x,y)\) approaches \((0,0)\) along the line \(y=mx\text{?}\)
(d)
Based on what you have seen so far, do you think \(\displaystyle\lim_{(x,y)\to(0,0)}g(x,y)\) exists? If so, what do you think its value is?
(e)
Now consider the behavior of \(g\) on the parabola \(y=x^2\text{?}\) What is \(g(x,x^2)\) and what is the limit of \(g\) as \((x,y)\) approaches \((0,0)\) along this parabola?
(f)
State whether you think the limit \(\displaystyle{\lim_{(x,y)\to(0,0)} g(x,y)}\) exists or not and provide a justification of your statement.
This activity shows that we need to be careful when studying the limit of a two-variable functions by considering its behavior along different paths. If we find two different paths that result in two different limits, then we conclude that the limit does not exist. However, we can never conclude that the limit of a function does exist only by considering its behavior along different paths.
Generally speaking, demonstrating that the limit \(\displaystyle{\lim_{(x,y)\to(a,b)}f(x,y)}\) exists requires a more careful argument.
Example 11.2.9.
We consider the function \(f\) defined by
\begin{equation*}
f(x,y) = \frac{x^2y^2}{x^2+y^2}\text{.}
\end{equation*}
We will try to determine whether \(\displaystyle{\lim_{(x,y)\to(0,0)}f(x,y)}\) exists or not.
Note that if either \(x\) or \(y\) is 0, then \(f(x,y) = 0\text{.}\) Therefore, if \(f\) has a limit at \((0,0)\text{,}\) it must be 0. We will therefore argue that
\begin{equation*}
\lim_{(x,y)\to(0,0)}f(x,y) = 0
\end{equation*}
by showing that we can make \(f(x,y)\) as close to \(0\) as we wish by taking \((x,y)\) sufficiently close (but not equal) to \((0,0)\text{.}\) In the following arguments, we view \(x\) and \(y\) as being real numbers that are close, but not equal, to 0.
Since \(0 \leq x^2\text{,}\) we have
\begin{equation*}
y^2 \leq x^2+y^2\text{,}
\end{equation*}
which implies that
\begin{equation*}
\frac{y^2}{x^2+y^2} \leq 1\text{.}
\end{equation*}
Multiplying both sides by \(x^2\) and observing that \(f(x,y) \ge 0\) for all \((x,y)\) gives
\begin{equation*}
x^2 \geq x^2\left(\frac{y^2}{x^2+y^2}\right) = \frac{x^2y^2}{x^2+y^2} = f(x,y) \geq 0\text{.}
\end{equation*}
Thus, \(0 \leq f(x,y) \leq x^2\text{.}\) Since \(x^2 \to 0\) as \(x \to 0\text{,}\) we can make \(f(x,y)\) as close to \(0\) as we like by taking \(x\) sufficiently close to \(0\text{.}\) For this example, it turns out that we don’t even need to worry about making \(y\) close to 0. Therefore,
\begin{equation*}
\lim_{(x,y) \to (0,0)} \frac{x^2y^2}{x^2+y^2} = 0\text{.}
\end{equation*}
In spite of the fact that these two most recent examples illustrate some of the complications that arise when studying limits of two-variable functions, many of the properties that are familiar from our study of single variable functions hold in precisely the same way.
Properties of Limits.
Let \(f=f(x,y)\) and \(g=g(x,y)\) be functions so that \(\displaystyle{\lim_{(x,y) \to (a,b)} f(x,y)}\) and \(\displaystyle{\lim_{(x,y) \to (a,b)} g(x,y)}\) both exist. Then
-
\(\displaystyle \lim_{(x,y) \to (a,b)} cf(x,y) = c\left(\lim_{(x,y) \to (a,b)} f(x,y)\right)\) for any scalar \(c\)
-
\(\displaystyle \displaystyle \lim_{(x,y) \to (a,b)} [f(x,y) \pm g(x,y)] = \lim_{(x,y) \to (a,b)} f(x,y) \pm \lim_{(x,y) \to (a,b)} g(x,y) \)
-
\(\displaystyle \displaystyle \lim_{(x,y) \to (a,b)} [f(x,y) g(x,y)] = \left(\lim_{(x,y) \to (a,b)} f(x,y)\right) \left( \lim_{(x,y) \to (a,b)} g(x,y)\right) \)
-
\(\displaystyle \lim_{(x,y) \to (a,b)} \frac{f(x,y)}{g(x,y)} = \frac{\displaystyle \lim_{(x,y) \to (a,b)} f(x,y)}{\displaystyle \lim_{(x,y) \to (a,b)} g(x,y)}\) if \(\displaystyle \lim_{(x,y) \to (a,b)} g(x,y) \neq 0\)
We can use these properties and results from single variable calculus to verify that many limits exist. For example, these properties show that the function \(f\) defined by
\begin{equation*}
f(x,y) = 3x^2y^3 + 2xy^2 - 3x + 1
\end{equation*}
has a limit at every point \((a,b)\) and, moreover,
\begin{equation*}
\lim_{(x,y)\to(a,b)} f(x,y) = f(a,b).
\end{equation*}
The reason for this is that polynomial functions of a single variable have limits at every point.
Subsection 11.2.3 Using Polar Coordinates in Limits
In Activity 11.2.3, 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, Example 11.2.9 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 Subsection 9.8.2 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 Subsection 9.8.2, there are several examples of converting points and equations into polar coordinates so will refer you there to review as necessary.
Example 11.2.10.
We want to understand the following limit, which we first saw in Activity 11.2.2, by using polar coordinates:
\begin{equation*}
\lim_{(x,y) \to (0,0)} \frac{y}{\sqrt{x^2+y^2}}
\end{equation*}
First, we note that this function is well-suited to convert to polar coordinates by writing \(x\) and \(y\) in terms of \(r\) and \(\theta\text{.}\) Substituting in \(y=r \sin(\theta)\) and \(r^2=x^2+y^2\) into our function we have
\begin{equation*}
\frac{y}{\sqrt{x^2+y^2}} = \frac{r\sin(\theta)}{\sqrt{r^2}} = \sin(\theta) \text{.}
\end{equation*}
Notice that this further simplifies to \(\sin(\theta)\text{.}\)
We now convert the limit expression from \((x,y) \to (0,0)\) to \(r \to 0\text{.}\) The whole conversion of the limit becomes
\begin{equation*}
\lim_{(x,y) \to (0,0)} \frac{y}{\sqrt{x^2+y^2}} = \lim_{r \to 0} \sin(\theta) \text{.}
\end{equation*}
Evaluating the limit of \(\sin(\theta)\) as \(r\) goes to \(0\) gives \(\sin(\theta)\text{.}\) The result of our algebraic limit calculation may seem surprising at first, because your prior experience with limits suggests that the limit of a function should either be a scalar or not exist. The conversion from rectangular to polar coordinates means we must interpret this result in terms of rectangular coordinates. Another way of saying what it means for the limit to be \(\sin(\theta)\) is that our conclusion is that the limit has a dependence on the direction in which we approached the origin. Specifically, when approaching the origin along a line that makes an angle of \(\theta\) with the positive \(x\)-axis, the output of the function approaches \(\sin(\theta)\text{.}\) In terms of rectangular coordinates, we say that the limit of \(\frac{y}{\sqrt{x^2+y^2}}\) as \((x,y)\to(0,0)\) does not exist because the limits along different paths going to the origin do not agree with each other.
Figure 11.2.11 shows in blue a collection of paths in the \(xy\)-plane that approach the origin. Each of these paths is shown in red on the surface \(z=f(x,y)=\frac{y}{\sqrt{x^2+y^2}}\) to demonstrate how the output of \(f\text{,}\) which is represented as the height of the surface, approaches different values as these paths each approach \((0,0)\text{.}\) Note that the limit along each of these paths exists, but the limit as \((x,y) \to (0,0)\) does not exist because there are paths that yield different limits. This should give you a greater appreciation for a surface plot like we saw in Activity 11.2.2. Computer-generated plots will create a surface plot but it is important for you to identify features on that plot that will have behaviour quite different than the smooth, continuous appearance.
Activity 11.2.4.
In this activity, we again consider the limit as \((x,y)\) goes to \((0,0)\) of \(g(x,y)=\frac{x^2y^2}{x^2+y^2}\text{,}\) which we previously saw in Example 11.2.9.
(a)
Convert \(g\) to polar coordinates.
(b)
Find the limit as \(r\to 0\) of your expression in polar coordinates.
(c)
Write a couple of sentences explaining how your answer to the previous part shows that the original limit is \(0\) and contrast with the previous example.
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 convienent it was to convert to polar coordinates in these functions. If we were examining a function that did not have a convienent 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 Exercise 13.) 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.
Subsection 11.2.4 Continuity
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:
-
\(f(a)\) exists,
-
\(\displaystyle{\lim_{x\to a}f(x)}\) exists, and
-
\(\displaystyle \displaystyle{\lim_{x\to a}f(x)=f(a)}\)
Using our understanding of limits of multivariable functions, we can define continuity in the same way.
Definition 11.2.12.
A function \(f=f(x,y)\) is continuous at the point \((a,
b)\) provided that
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\(\displaystyle{\lim_{(x,y) \to (a,b)} f(x,y)}\) exists, and
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\(\displaystyle{\lim_{(x,y) \to (a,b)} f(x,y) = f(a,b)}\text{.}\)
For instance, we have seen that the function \(f\) defined by \(f(x,y) = 3x^2y^3 + 2xy^2 - 3x + 1\) is continous at every point. And just as with single variable functions, continuity has certain properties that are based on the properties of limits.
Properties of continuity.
Let \(f\) and \(g\) be functions of two variables that are continuous at the point \((a,b)\text{.}\) Then
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.
Summary
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A function \(f = f(x,y)\) has a limit \(L\) at a point \((a,b)\) provided that we can make \(f(x,y)\) as close to \(L\) as we like by taking \((x,y)\) sufficiently close (but not equal) to \((a,b)\text{.}\)
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If \((x,y)\) has two different limits as \((x,y)\) approaches \((a,b)\) along two different paths, we can conclude that \(\lim_{(x,y)\to(a,b)}f(x,y)\) does not exist.
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Properties similar to those for one-variable functions allow us to conclude that many limits exist and to evaluate them.
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A function \(f = f(x,y)\) is continuous at a point \((a,b)\) in its domain if \(f\) has a limit at \((a,b)\) and\begin{equation*} f(a,b) = \lim_{(x,y) \to (a,b)} f(x,y). \end{equation*}
Exercises 11.2.5 Exercises
1.
2. Determining the limit of a function.
In this problem we show that the function
\begin{equation*}
f(x,y) = \frac{2x^{2}-y^{2}}{x^{2}+y^{2}}
\end{equation*}
does not have a limit as \((x,y)\to (0,0)\text{.}\)
(a) Suppose that we consider \((x,y)\to (0,0)\) along the curve \(y = 3 x\text{.}\) Find the limit in this case:
\(\lim\limits_{(x,3 x)\to(0,0)} \frac{2x^{2}-y^{2}}{x^{2}+y^{2}} =\)
(b) Now consider \((x,y)\to (0,0)\) along the curve \(y = 4 x\text{.}\) Find the limit in this case:
\(\lim\limits_{(x,4 x)\to(0,0)} \frac{2x^{2}-y^{2}}{x^{2}+y^{2}} =\)
(c) Note that the results from (a) and (b) indicate that \(f\) has no limit as \((x,y)\to (0,0)\) (be sure you can explain why!).
To show this more generally, consider \((x,y)\to (0,0)\) along the curve \(y = m x\text{,}\) for arbitrary \(m\text{.}\) Find the limit in this case:
\(\lim\limits_{(x,m x)\to(0,0)} \frac{2x^{2}-y^{2}}{x^{2}+y^{2}} =\)
(Be sure that you can explain how this result also indicates that \(f\) has no limit as \((x,y)\to(0,0)\text{.}\)
3.
Show that the function
\begin{equation*}
f(x,y) = \frac{x^{3}y}{x^{6}+y^{3}}.
\end{equation*}
does not have a limit at \((0,0)\) by examining the following limits.
\(\lim\limits_{{}^{(x,y)\to(0,0)}_{y=x}} f(x,y) =\)
\(\lim\limits_{{}^{(x,y)\to(0,0)}_{y=x^3}} f(x,y) =\)
(Be sure that you are able to explain why the results in (a) and (b) indicate that \(f\) does not have a limit at (0,0)!
4.
5.
6.
7.
8.
9.
10.
The largest set on which the function \(f(x,y) = 1/(3 - x^2 - y^2)\) is continuous is
11.
Consider the function \(f\) defined by \(f(x,y) = \frac{xy}{x^2 + y^2 + 1}.\)
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What is the domain of \(f\text{?}\)
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Evaluate limit of \(f\) at \((0,0)\) along the following paths: \(x = 0\text{,}\) \(y = 0\text{,}\) \(y = x\text{,}\) and \(y = x^2\text{.}\)
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What do you conjecture is the value of \(\lim_{(x,y) \to (0,0)} f(x,y)\text{?}\)
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Use appropriate technology to sketch both surface and contour plots of \(f\) near \((0,0)\text{.}\) Write several sentences to say how your plots affirm your findings in (a) - (d).
12.
Consider the function \(g\) defined by \(g(x,y) = \frac{xy}{x^2 + y^2}.\)
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What is the domain of \(g\text{?}\)
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Evaluate limit of \(g\) at \((0,0)\) along the following paths: \(x = 0\text{,}\) \(y = x\text{,}\) and \(y = 2x\text{.}\)
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What can you now say about the value of \(\lim_{(x,y) \to (0,0)} g(x,y)\text{?}\)
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Use appropriate technology to sketch both surface and contour plots of \(g\) near \((0,0)\text{.}\) Write several sentences to say how your plots affirm your findings in (a) - (d).
13.
Consider the function \(h\) defined by \(h(x,y) = \frac{2x^2y}{x^4 + y^2}.\)
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What is the domain of \(h\text{?}\)
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Evaluate the limit of \(h\) at \((0,0)\) along all linear paths the contain the origin. What does this tell us about \(\lim_{(x,y) \to (0,0)} h(x,y)\text{?}\) (Hint: A non-vertical line throught the origin has the form \(y = mx \) for some constant \(m\text{.}\))
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Does \(\lim_{(x,y) \to (0,0)} h(x,y)\) exist? Verify your answer. Check by using appropriate technology to sketch both surface and contour plots of \(h\) near \((0,0)\text{.}\) Write several sentences to say how your plots affirm your findings about \(\lim_{(x,y) \to (0,0)} h(x,y)\text{.}\)
14.
For each of the following prompts, provide an example of a function of two variables with the desired properties (with justification), or explain why such a function does not exist.
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A function \(p\) that is defined at \((0,0)\text{,}\) but \(\lim_{(x,y) \to (0,0)} p(x,y)\) does not exist.
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A function \(q\) that does not have a limit at \((0,0)\text{,}\) but that has the same limiting value along any line \(y = mx\) as \(x \to 0\text{.}\)
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A function \(r\) that is continuous at \((0,0)\text{,}\) but \(\lim_{(x,y) \to (0,0)} r(x,y)\) does not exist.
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A function \(s\) such that\begin{equation*} \lim_{(x,x) \to (0,0)} s(x,x) = 3 \ \ \ \mbox{and} \ \ \ \lim_{(x,2x) \to (0,0)} s(x,2x) = 6, \end{equation*}for which \(\lim_{(x,y) \to (0,0)} s(x,y)\) exists.
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A function \(t\) that is not defined at \((1,1)\) but \(\lim_{(x,y) \to (1,1)} t(x,y)\) does exist.
15.
Use the properties of continuity to determine the set of points at which each of the following functions is continuous. Justify your answers.
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The function \(h\) defined by\begin{equation*} h(x,y) = \begin{cases} \frac{xy}{x^2+y^2} \amp \text{ if } (x,y) \neq (0,0) \\ 0 \amp \text{ if } (x,y) = (0,0) \end{cases} \end{equation*}
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The function \(k\) defined by\begin{equation*} k(x,y) = \begin{cases} \frac{x^2y^4}{x^2+y^2} \amp \text{ if } (x,y) \neq (0,0) \\ 0 \amp \text{ if } (x,y) = (0,0) \end{cases} \end{equation*}
