Subsection 4.2.1 Limits
To make sense of a statement like “\(f(x,y)\) has limit \(L\) as the arguments approach the point \((a,b)\)” we need to quantify the idea of a pair of arguments \(x\) and \(y\) being close to another pair \(a\) and \(b\text{.}\) This is done with the distance between the points \((x,y)\) and \((a,b)\) in the plane.
Thus the precise
\(\delta-\epsilon\) definition of a limit is
Definition 4.2.1.
\(f(x,y)\) has limit \(L\) as \((x,y)\) approaches \((a,b)\) if for any positive value \(\epsilon\text{,}\) there is a positive value \(\delta\) such that for any point \((x,y)\) in the domain \(D\) of \(f\) (other than \((a,b)\) itself), having \(0 \lt \sqrt{(x-a)^2+(y-b)^2} \lt \delta\) ensures that \(|f(x,y)-L| \lt \epsilon\text{.}\)
Note that only points \((x,y) \neq (a,b)\) matter.
If the limit exists and has value \(L\text{,}\) we write
\begin{equation*}
\lim_{(x,y)\to(a,b)} f(x,y) = L.
\end{equation*}
It is sometimes nice to consider \(f\) as a function of a vector argument \(\vec{x}=\vector{x,y}\) and the point \((a,b)\) as the vector \(\vector{a,b}\text{.}\)
Then the condition is that for any \(\vec{x} = \vector{x,y}\) in domain \(D\) other than \(\vec{a} = \vector{a,b}\text{,}\)
\begin{equation*}
\|\vec{x}-\vec{a}\| \lt \delta \text{ ensures that } |f(\vec{x})-L| \lt \epsilon\text{.}
\end{equation*}
One advantage of this is that it applies equally well for functions on three variables (or even more). In fact it also works for functions on the real numbers: just take \(\| \vec{x}-\vec{a} \|\) to mean \(| x-a |\) and so on.
Limit Laws.
All the familiar limits laws apply, for sums, differences, products, quotients and so on.
Some Cautionary Examples: Limits can Fail to Exist in New Ways.
With functions of one variable, it is sometimes convenient to compute limits using the one-sided limits from each side.
It might seem worth trying a similar thing with functions of two variables: approach a point \((a,b)\) along various straight lines passing through it, so that one is really working with a function of one variable: \(g(t) = f(a+ct,b+dt)\) with \(\vector{c,d}\) giving the direction of the line.
However, strange things can happen:
Exercise 4.2.2.
For the functions \(f\) and \(g\) given by
\begin{equation*}
f(x,y) = \frac{\sin(x^2+y^2)}{x^2+y^2} \quad \text{ and } \quad \frac{x^2-y^2}{x^2+y^2}
\end{equation*}
with domain everything except the origin \((0,0)\text{,}\) consider how the values behave as \((x,y)\) approaches the origin along straight lines.
For each function, does the limit exist as \((x,y) \to (a,b)\) and if so, what is it?
Hint.
Parameterize the lines as \(\vector{at, bt}\text{,}\) where \(\vec{v} = \vector{a,b}\) is a [unit] vector describing the direction of approach.
Exercise 4.2.3.
For the function
\begin{equation*}
f(x, y) = \left\{ \begin{array}{rl}
\displaystyle \frac{x^2}{y}, \amp y \neq 0 \\
0, \amp y = 0
\end{array} \right.
\end{equation*}
Show that approaching the origin on any straight line gives \(f(x,y) \to 0\text{,}\) but …
… it does not have a limit at the origin.
Hint.
Look at the curves \(y=kx^2\)
Interior Points and Boundary Points.
For functions of one variable, the domain is typically an interval like \((a, b)\text{,}\) \([a, b]\text{,}\) \((a, b]\) or \([a, b)\) (or a union of these), and in the latter three cases, the domain includes endpoints, \(a\) and/or \(b\) where limits are handled a bit differently, as one-sided limits.
The issue of the limit at a boundary point \(\vec{a}\) is now dealt with more elegantly by simply requiring that \(|f(\vec{x}) - L \lt \epsilon\) for any \(\vec{x}\) in the domain of \(f\) where \(\| \vec{x} - \vec{a} \| \lt \delta\text{.}\) That is, we just ignore points \(\vec{x}\) outside the domain.
Nevertheless, it is useful to consider what happens at the "edges" of a domain in two (or more) dimensions. The generalization of endpoints is the
boundary of a set
\(D\) in
\(\reals^2\) or
\(\reals^3\text{,}\) as contrasted to the
interior:
Definition 4.2.4.
A point \(\vec{a}\) in any set \(D\) is in the interior of \(D\) if it is surrounded by a small enough "disk" or "ball" that lies entirely in \(D\text{.}\) That is, there is some radius \(\delta > 0\) such that all points in the set \(B_\delta(\vec{a}) = \{ \vec{x}: \| \vec{x} - \vec{a} \| \lt \delta \}\) of radius \(\delta\) and center \(\vec{a}\) are also in \(D\text{.}\)
Any such set \(B_\delta(\vec{x})\) is called a ball, even though in two dimensions it is a disk. This jargon is sometimes even used in one dimension, where a "ball" is an open interval \((a-\delta, a+\delta\)).
The points of \(D\) that are not in its interior are on its boundary. However, the boundary includes points that are "adjacent" to \(D\) even if they are not in it: for example the boundary of \(B_\delta(\vec{x})\) is the circle in \(\reals^2\) (or sphere in \(\reals^3\)) where \(\| \vec{x} - \vec{a} \| = \delta\text{,}\) and this is entirely outside that ball.
Definition 4.2.5.
A point \(\vec{a}\) is in the boundary of \(D\) if no ball \(B_\delta(\vec{a})\) is either entirely in \(D\) or entirely outside it.
That is, every ball around a point in the boundary is partly in the set, partly outside it.
Yet another way of saying this is that one can get as close as you want to such a boundary point both from inside the set and from outside it.
The boundary of a set \(D\) is sometimes denoted \(\partial D\text{.}\)
Exercise 4.2.6.
Describe the boundaries of each of the four sets \((a, b)\text{,}\) \([a, b]\text{,}\)\([a, b)\) and \((a, b]\text{?}\)
Open and Closed Sets.
There are two opposite extremes that can make a domain easier to work with:
Definition 4.2.7.
A set is open if it consists entirely of interior points: it contains none of its boundary points.
A familiar example is open intervals like \((a, b)\text{;}\) the balls \(B_\delta(\vec{a})\) defined above are another example: hence these are sometimes called open balls.
Definition 4.2.8.
A set is closed if it contains all of its boundary points.
A familiar example is closed intervals like \([a, b]\text{.}\) There are also the closed balls
\begin{equation*}
\bar{B}_\delta(\vec{a}) = \{ \vec{x}: \| \vec{x} - \vec{a} \| \le \delta \}
\end{equation*}
(Pay close attention to the inequality sign!)
Subsection 4.2.2 Continuity
Continuity is now defined in the familiar way:
Definition 4.2.10.
a function \(f\) is continuous at \((a,b)\) if
\begin{equation*}
\lim_{(x,y)\to(a,b)}f(x,y) = f(a,b),
\end{equation*}
and \(f\) is continuous if it is continuous at each point \((a,b)\) in its domain.
The familar properties of continuity apply, for sums, differences, products, compositions (so long as the range of the "inner" function is within the domain of the "outer" one) and quotients (where they are defined; away from division by zero).
Polynomials and Rational Functions.
Definition 4.2.11.
A polynomial function in two variables \(x\) and \(y\) is any sum of constant multiples of terms like \(x^n y^m\) with \(m\) and \(n\) non-negative integers, and a rational function in two variables is a quotient of such polynomials.
As one might guess, polynomials in one variable are continuous at every point, and so are simply continuous, and rational functions are continuous at any point where the denominator is non-zero.
But more is true: since points with zero denominator are not in the domain of the rational function, rational functions are also continuous!
Subsection 4.2.3 Functions of Three or More Variables
All the above ideas of limits, continuity, polynomials and rational functions extend in an obvious way to functions of three or more variables, like \(f(x,y,z)\) or \(f(x_1, x_2, \dots,x_n)\text{.}\)
The vector notation \(\vec{x}=\vector{x,y}\text{,}\) \(\vec{x}=\vector{x,y,z}\) or \(\vec{x}=\vector{x_1, x_2, \dots,x_n}\) becomes convenient for dealing with functions of any number of independent variables by using the common notation \(f(\vec{x})\text{.}\)
We have:
Definition 4.2.12.
Function \(f\) with domain \(D\) in \(\mathbb{R}^n\) has limit \(L\) as \(\vec{x}\) approaches \(\vec{a}\) if for any positive \(\epsilon\text{,}\) there is a positive \(\delta\) such that
\begin{equation*}
|f(\vec{x})-L| \lt \epsilon \text{ whenever } 0 \lt \|\vec{x}-\vec{a}\| \lt \delta \text{ and } \vec{x}
\text{ is in }D.
\end{equation*}
(Note that only points \(\vec{x} \neq \vec{a}\) matter.)
If so, we write
\begin{equation*}
\lim_{\vec{x}\to\vec{a}}f(\vec{x})=L.
\end{equation*}
