<< Chapter < Page Chapter >> Page >
  • Calculate the limit of a function of two variables.
  • Learn how a function of two variables can approach different values at a boundary point, depending on the path of approach.
  • State the conditions for continuity of a function of two variables.
  • Verify the continuity of a function of two variables at a point.
  • Calculate the limit of a function of three or more variables and verify the continuity of the function at a point.

We have now examined functions of more than one variable and seen how to graph them. In this section, we see how to take the limit of a function of more than one variable, and what it means for a function of more than one variable to be continuous at a point in its domain. It turns out these concepts have aspects that just don’t occur with functions of one variable.

Limit of a function of two variables

Recall from Section 2.2 the definition of a limit of a function of one variable:

Let f ( x ) be defined for all x a in an open interval containing a . Let L be a real number. Then

lim x a f ( x ) = L

if for every ε > 0 , there exists a δ > 0 , such that if 0 < | x a | < δ for all x in the domain of f , then

| f ( x ) L | > ε .

Before we can adapt this definition to define a limit of a function of two variables, we first need to see how to extend the idea of an open interval in one variable to an open interval in two variables.

Definition

Consider a point ( a , b ) 2 . A δ disk    centered at point ( a , b ) is defined to be an open disk of radius δ centered at point ( a , b ) —that is,

{ ( x , y ) 2 | ( x a ) 2 + ( y b ) 2 < δ 2 }

as shown in the following graph.

On the xy plane, the point (2, 1) is shown, which is the center of a circle of radius δ.
A δ disk centered around the point ( 2 , 1 ) .

The idea of a δ disk appears in the definition of the limit of a function of two variables. If δ is small, then all the points ( x , y ) in the δ disk are close to ( a , b ) . This is completely analogous to x being close to a in the definition of a limit of a function of one variable. In one dimension, we express this restriction as

a δ < x < a + δ .

In more than one dimension, we use a δ disk.

Definition

Let f be a function of two variables, x and y . The limit of f ( x , y ) as ( x , y ) approaches ( a , b ) is L , written

lim ( x , y ) ( a , b ) f ( x , y ) = L

if for each ε > 0 there exists a small enough δ > 0 such that for all points ( x , y ) in a δ disk around ( a , b ) , except possibly for ( a , b ) itself, the value of f ( x , y ) is no more than ε away from L ( [link] ). Using symbols, we write the following: For any ε > 0 , there exists a number δ > 0 such that

| f ( x , y ) L | < ε whenever 0 < ( x a ) 2 + ( y b ) 2 < δ .
In xyz space, a function is drawn with point L. This point L is the center of a circle of radius ॉ, with points L ± ॉ marked. On the xy plane, there is a point (a, b) drawn with a circle of radius δ around it. This is denoted the δ-disk. There are dashed lines up from the δ-disk to make a disk on the function, which is called the image of delta disk. Then there are dashed lines from this disk to the circle around the point L, which is called the ॉ-neighborhood of L.
The limit of a function involving two variables requires that f ( x , y ) be within ε of L whenever ( x , y ) is within δ of ( a , b ) . The smaller the value of ε , the smaller the value of δ .

Proving that a limit exists using the definition of a limit of a function of two variables can be challenging. Instead, we use the following theorem, which gives us shortcuts to finding limits. The formulas in this theorem are an extension of the formulas in the limit laws theorem in The Limit Laws .

Limit laws for functions of two variables

Let f ( x , y ) and g ( x , y ) be defined for all ( x , y ) ( a , b ) in a neighborhood around ( a , b ) , and assume the neighborhood is contained completely inside the domain of f . Assume that L and M are real numbers such that lim ( x , y ) ( a , b ) f ( x , y ) = L and lim ( x , y ) ( a , b ) g ( x , y ) = M , and let c be a constant. Then each of the following statements holds:

Constant Law:

lim ( x , y ) ( a , b ) c = c

Identity Laws:

lim ( x , y ) ( a , b ) x = a
lim ( x , y ) ( a , b ) y = b

Sum Law:

lim ( x , y ) ( a , b ) ( f ( x , y ) + g ( x , y ) ) = L + M

Difference Law:

lim ( x , y ) ( a , b ) ( f ( x , y ) g ( x , y ) ) = L M

Constant Multiple Law:

lim ( x , y ) ( a , b ) ( c f ( x , y ) ) = c L

Product Law:

lim ( x , y ) ( a , b ) ( f ( x , y ) g ( x , y ) ) = L M

Quotient Law:

lim ( x , y ) ( a , b ) f ( x , y ) g ( x , y ) = L M for M 0

Power Law:

lim ( x , y ) ( a , b ) ( f ( x , y ) ) n = L n

for any positive integer n .

Root Law:

lim ( x , y ) ( a , b ) f ( x , y ) n = L n

for all L if n is odd and positive, and for L 0 if n is even and positive.

Practice Key Terms 8

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Source:  OpenStax, Calculus volume 3. OpenStax CNX. Feb 05, 2016 Download for free at http://legacy.cnx.org/content/col11966/1.2
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