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 begin to understand 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.
1
Remember the steps to the classic calculus approach: 1) approximate the measurement, 2) quantify how the approximation changes on a finer scale, and 3) use a limit to show how the approximation converges to the measurement of interest
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. The limit of a of function like \(\vr(t)=\langle x(t),y(t),z(t) \rangle\) is evaluated as three separate one variable limits.
Let’s 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
\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 relook at several ideas related to limits of (single variable) functions by working with tables, graphs, and the algebraic forms of these functions.
(a)
Let \(f\) defined by \(f(x) = 3-x\text{.}\) Complete Table 11.2.1. Table 11.2.1. Values of \(f(x) = 3-x\text{.}\)
| \(x\) | \(-0.2\) | \(-0.1\) | \(-0.01\) | \(0.01\) | \(0.1\) | \(0.2\) |
| \(f(x)\) |
(b)
Write a sentence about what Table 11.2.1 suggests regarding
\begin{equation*}
\lim_{x\to 0}f(x)
\end{equation*}
(c)
Write a couple of sentences to explain how your values in Table 11.2.1 and the
\begin{equation*}
\lim_{x\to 0}f(x)
\end{equation*}
are demonstrated in Figure 11.2.4.
(d)
Let
\begin{equation*}
g(x) = \frac{x}{|x|}\text{.}
\end{equation*}
Fill in the blanks of Table 11.2.5 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 Table 11.2.5 suggests about \(\lim_{x \to 0}g(x)\text{.}\)
(f)
Write a couple of sentences to explain how your values in Table 11.2.5 and the
\begin{equation*}
\lim_{x\to 0} g(x)
\end{equation*}
are demonstrated in Figure 11.2.8.
(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 tasks.
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, the limit \(\displaystyle{\lim_{x \rightarrow a} f(x)}\) (if it exists) will be the value that the output of \(f\) approaches as the inputs (\(x\)-values) get closer to \(a\text{.}\) We must look at whether the output of \(f\) approaches the same value for 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.
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 that kind of generalization involves a lot of mathematical tools that are not appropriate at this time.
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, as shown in Figure 11.2.9. When considering a limit of a single variable function, you only have to evaluate and compare the output of your function in these two directions. If the output of your function did not agree as you approach from the left and right, then the limit did not exist (like 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 A LOT of different ways to approach a point like \((a,b)\text{,}\) the input for a function like \(f(x,y)\text{.}\) In Figure 11.2.10, you can see a plot of seven different ways to approach the point \((a,b)\text{.}\) When we look at how to get close to a input in two dimensions, we will need to think about more than just the straight-line directions; we have to consider curved paths and other collections of points that aren’t even a continuous path. 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.
A plot of two vectors with an acute angle between them
Example 11.2.11.
In this example, we will examine the same tools used in Preview Activity 11.2.1 (tables and graphs) to think about what the output of the function \(f(x,y)=3-x-2y\) approaches as the inputs get close to \((0,0)\text{.}\) Table 11.2.12 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 11.2.12 shows that as the inputs, \((x,y)\text{,}\) get closer to \((0,0)\) 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 the same \(x\)- 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 11.2.13 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 in-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 very 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 quite straightforward because the surface is a plane. We can see in Figure 11.2.14 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) will get closer to 3.
If you were not convinced that the output of \(f\) approaches 3 as \((x,y) \to (0,0)\text{,}\) 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, but rather only at points of the form \((x,2x)\text{.}\) Our limit expression becomes a 1-dimensional problem in this case because along the path given by \(y=2x\)
\begin{align*}
(\text{along }y=2x) \quad \lim_{(x,y)\to (0,0)} f(x,y) \amp\Rightarrow \\
\quad \quad \lim_{(x,2x)\to (0,0)} f(x,2x) \amp= \lim_{x \to 0} 3-x-2(2x)
\end{align*}
Note that the limit along this path is now something like what you evaluated in your single variable calculus work. In fact, you can evaluate \(\displaystyle{\lim_{x \to 0} 3-x-2(2x)}\) to get 3.
This idea for reducing a two variable limit to a one variable limit is very versatile, but has a significant drawback. If you wanted to approach the origin along the path given by \(x=sin(y)\) or along the path parameterized by \(\vr(t)= \langle t \cos(t) , t \sin(t) \rangle\text{,}\) then you would get the following:
\begin{align*}
(\text{along }x=\sin(y)) \quad\amp \lim_{(x,y)\to (0,0)} f(x,y)\Rightarrow \\
\lim_{(\sin(y),y)\to (0,0)} f(\sin(y),y) \amp= \lim_{y \to 0} 3-\sin(y)-2(y) \\
\amp= 3-0-0 = 3
\end{align*}
\begin{align*}
(\text{along }\vr(t)= \langle t \cos(t) , t \sin(t) \rangle) \quad\amp \lim_{(x,y)\to (0,0)} f(x,y) \Rightarrow \\
\quad \quad \lim_{t \to 0} f(t \cos(t), t \sin(t)) \amp= \lim_{t \to 0} 3-t \cos(t)-2(t \sin(t)) \\
\amp= 3-0-0 = 3
\end{align*}
Remember that you will never be able to test all possible paths that approach this limit point, so while this idea to reduce down to a particular path is 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 11.2.15.
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 11.2.11 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}}
\end{equation*}
whose graph is shown below in Figure 11.2.16
(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)
Values of \(f\) (to three decimal places) at several points close to \((0,0)\) are shown in Table 11.2.17.
| \(x\backslash y\) | \(-1.000\) | \(-0.100\) | \(0.000\) | \(0.100\) | \(1.000\) |
| \(-1.000\) | \(-0.707\) | — | \(0.000\) | — | \(0.707\) |
| \(-0.100\) | — | \(-0.707\) | \(0.000\) | \(0.707\) | — |
| \(0.000\) | \(-1.000\) | \(-1.000\) | — | \(1.000\) | \(1.000\) |
| \(0.100\) | — | \(-0.707\) | \(0.000\) | \(0.707\) | — |
| \(1.000\) | \(-0.707\) | — | \(0.000\) | — | \(0.707\) |
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, \(f(x,0)= \text{?}\)
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 (thus considering \(f(x,x)\) as \(x \to 0^+\)), 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 justify your idea.
(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)
\end{equation*}
Use this limit to explain the the behavior of \(f(x,y)\) as \((x,y)\) approaches \((0,0)\) along line 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 11.2.18; \(\displaystyle{\lim_{x \to 0}g(x)}\) does not exist because we see different limits as \(x\) approaches 0 from the left and the right.
As a general rule, we have
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.
Let’s consider the function \(g\) defined by
\begin{equation*}
g(x,y) = \frac{x^2y}{x^4 + y^2}
\end{equation*}
and 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 \(\lim_{(x,y)\to(0,0)}g(x,y)\) 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 may conclude that the limit does not exist. However, we can never conclude that the limit of a function exists only by considering its behavior along different paths.
Generally speaking, demonstrating that a limit, \(\displaystyle{\lim_{(x,y)\to(a,b)}f(x,y)}\text{,}\) exists requires a more careful argument.
Example 11.2.19.
We will consider the function \(f\) defined by
\begin{equation*}
f(x,y) = \frac{x^2y^2}{x^2+y^2}.
\end{equation*}
and 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
\end{equation*}
which implies that
\begin{equation*}
\frac{y^2}{x^2+y^2} \leq 1
\end{equation*}
Multiplying both sides by \(x^2\) and observing that \(f(x,y) \ge 0\) for all \((x,y)\) gives
\begin{equation*}
0\leq f(x,y) = \frac{x^2y^2}{x^2+y^2} = x^2\left(\frac{y^2}{x^2+y^2}\right) \leq x^2
\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
\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 11.2.19 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 11.2.20.
In this example, we will look at how polar coordinates can give some additional algebraic tools with respect to the functions explored in Activity 11.2.2 and Example 11.2.19.
(a)
For this part, we will look at converting the limit from Activity 11.2.2 to polar coordinates and seeing how this algebra will give the same determination of our limit. In particular, we want to understand the following limit:
\begin{equation*}
\lim_{(x,y) \to (0,0)} \frac{y}{\sqrt{x^2+y^2}}
\end{equation*}
First, we will convert our function to polar coordinates, which is pretty straighforward because we can write \(x\) and \(y\) in terms of \(r\) and \(\theta\text{.}\) Substituting in \(x=r \cos(\theta)\) and \(y=r \sin(\theta)\) (as well as \(r^2=x^2+y^2\)) into our function we get:
\begin{equation*}
\frac{y}{\sqrt{x^2+y^2}} = \frac{r\sin(\theta)}{\sqrt{r^2}} = \sin(\theta)
\end{equation*}
Note that we can can simplify our expression 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 looks like
\begin{equation*}
\lim_{(x,y) \to (0,0)} \frac{y}{\sqrt{x^2+y^2}} = \lim_{r \to 0} \sin(\theta)
\end{equation*}
Evaluating the limit of \(\sin(\theta)\) as \(r\) goes to 0 will give a result of \(\sin(\theta)\text{.}\) The result of our algebraic limit calculation may seem weird to you because the limit of a function should either be a scalar or not exist. The conversion from rectangular to polar coordinates means you must to interpret your result in terms of rectangular coordinates. We found that the limit of our function had a dependence on the direction in which we approached the origin. Specifically, if you approach the origin along an angle \(\theta\text{,}\) then the output of our function will approach \(\sin(\theta)\text{.}\) In terms of rectangular coordinates, we say that the limit of \(\frac{y}{\sqrt{x^2+y^2}}\) does not exist because the limits along different paths going to the origin do not agree with each other.
Figure 11.2.21 shows a collection of paths in the \(xy\)-plane that approach the origin (shown in blue). Each of these paths is shown on the surface given by \(z=f(x,y)=\frac{y}{\sqrt{x^2+y^2}}\) in red to demonstrate how the height/output of \(f\) will approach different values as these paths each approach \((0,0)\text{.}\) Note that the limit along each of these paths exists, but because the values of the limit along every path is not the same, the limit as \((x,y) \to (0,0)\) does not exist. This should give you a greater appreciation for a surface plot like Figure 11.2.16. 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.
(b)
In this task, we want to look at the limit as \((x,y)\) goes to \((0,0)\) of \(g(x,y)=\frac{x^2y^2}{x^2+y^2}\) as in Example 11.2.19. Converting \(g\) to polar coordinates gives
\begin{equation*}
\frac{x^2y^2}{x^2+y^2}=\frac{r^4 (\cos(\theta)\sin(\theta))^2}{r^2}=r^2 (\cos(\theta)\sin(\theta))^2
\end{equation*}
When we convert the limit \((x,y)\) goes to \((0,0)\) to the limit as \(r\) goes to 0 and evaluate, we get the following result.
\begin{equation*}
\lim_{(x,y) \to (0,0) } \frac{x^2y^2}{x^2+y^2}= \lim_{r \to 0} r^2 (\cos(\theta)\sin(\theta))^2 = 0
\end{equation*}
So as we approach the origin (from any direction), the output of \(g\) will approach 0. You can see how the different dependence on \(r\) of the output of our function causes there to be a different result on the limit (as compared to part (a) of this example). As we showed in Example 11.2.19,
\begin{equation*}
\lim_{(x,y) \to (0,0) } \frac{x^2y^2}{x^2+y^2}= 0
\end{equation*}
but the polar coordinate approach presented here likely helps you see where the argument used in Example 11.2.19 came from.
These examples may seem like the only tool you need is to convert to polar coordinates but that simplification misses how convienent the conversions to polar coordinates were for the expressions used in Example 11.2.20. If you were examining an expression that did not have a convienent conversion to polar coordinates or the algebra to simplify and evaluate your limit in polar coordinates was difficult, then polar coordinates will not be a great tool for evaluating limits. Polar coordinates as a tool is presented here because it highlights how to geometrically separate your measurements in two dimensions into a distance and a measure of directionality. These ideas come up 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 11.2.22.
A function \(f=f(x,y)\) is continuous at the point \((a,
b)\) provided that
-
\(\displaystyle{\lim_{(x,y) \to (a,b)} f(x,y)}\) exists, and
-
\(\displaystyle{\lim_{(x,y) \to (a,b)} f(x,y) = f(a,b)}\text{.}\)
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.
Subsection 11.2.5 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 \((a,b)\text{.}\)
-
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.
-
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 11.2.6 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 \(f(x,y) = \frac{xy}{x^2 + y^2 + 1}.\)
-
What is the domain of \(f\text{?}\)
-
Evaluate limit of \(f\) at \((0,0)\) along the following paths: \(x = 0\text{,}\) \(y = 0\text{,}\) \(y = x\text{,}\) and \(y = x^2\text{.}\)
-
What do you conjecture is the value of \(\lim_{(x,y) \to (0,0)} f(x,y)\text{?}\)
-
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 \(g(x,y) = \frac{xy}{x^2 + y^2}.\)
-
What is the domain of \(g\text{?}\)
-
Evaluate limit of \(g\) at \((0,0)\) along the following paths: \(x = 0\text{,}\) \(y = x\text{,}\) and \(y = 2x\text{.}\)
-
What can you now say about the value of \(\lim_{(x,y) \to (0,0)} g(x,y)\text{?}\)
-
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 \(h(x,y) = \frac{2x^2y}{x^4 + y^2}.\)
-
What is the domain of \(h\text{?}\)
-
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 \(m\text{.}\))
-
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.
-
A function \(p\) that is defined at \((0,0)\text{,}\) but \(\lim_{(x,y) \to (0,0)} p(x,y)\) does not exist.
-
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 \(x \to 0\text{.}\)
-
A function \(r\) that is continuous at \((0,0)\text{,}\) but \(\lim_{(x,y) \to (0,0)} r(x,y)\) does not exist.
-
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.
-
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.
-
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*}
-
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*}
