**14.2** Limits and Continuity

### Limits and Continuity

Let’s compare the behavior of the functions

and

*as x and y both approach 0 [and therefore the point (x, y) *
approaches the origin].

### Limits and Continuity

*Tables 1 and 2 show values of f(x, y) and g(x, y), correct to *
*three decimal places, for points (x, y) near the origin. *

(Notice that neither function is defined at the origin.)

### Limits and Continuity

**Table 2**

*Values of g(x, y)*

### Limits and Continuity

*It appears that as (x, y) approaches (0, 0), the values of *
*f(x, y) are approaching 1 whereas the values of g(x, y) *

aren’t approaching any number. It turns out that these

guesses based on numerical evidence are correct, and we write

and

does not exist

### Limits and Continuity

In general, we use the notation

*to indicate that the values of f(x, y) approach the number L *
*as the point (x, y) approaches the point (a, b) along any *
*path that stays within the domain of f.*

### Limits and Continuity

*In other words, we can make the values of f(x, y) as close *
*to L as we like by taking the point (x, y) sufficiently close to *
*the point (a, b), but not equal to (a, b). A more precise *

definition follows.

### Limits and Continuity

Other notations for the limit in Definition 1 are and

*f(x, y) → L as (x, y) → (a, b)*

*For functions of a single variable, when we let x approach a, *
there are only two possible directions of approach, from the
left or from the right.

We know that if lim_{x→a-}*f(x) ≠ lim*_{x→a+}*f(x), then lim*_{x→a}*f(x) *
does not exist.

### Limits and Continuity

For functions of two variables the situation is not as simple
*because we can let (x, y) approach (a, b) from an infinite *
number of directions in any manner whatsoever

*(see Figure 3) as long as (x, y) stays within the domain of f.*

**Figure 3**

### Limits and Continuity

*Definition 1 says that the distance between f(x, y) and L *
can be made arbitrarily small by making the distance from
*(x, y) to (a, b) sufficiently small (but not 0).*

*The definition refers only to the distance between *
*(x, y) and (a, b). It does not refer to the direction of *

approach.

*Therefore, if the limit exists, then f(x, y) must approach the *
*same limit no matter how (x, y) approaches (a, b).*

### Limits and Continuity

Thus, if we can find two different paths of approach along
*which the function f(x, y) has different limits, then it follows *
that lim*(x, y) → (a, b)* *f(x, y) does not exist.*

### Example 1

Show that does not exist.

Solution:

*Let f(x, y) = (x*^{2} *– y*^{2}*)/(x*^{2} *+ y*^{2}).

*First let’s approach (0, 0) along the x-axis. *

*Then y = 0 gives f(x, 0) = x*^{2}*/x*^{2} *= 1 for all x* ≠ 0, so

*f(x, y) → 1 as (x, y) → (0, 0) along the x-axis*

*Example 1 – Solution*

*We now approach along the y-axis by putting x = 0.*

*Then for all y* ≠ 0, so

*f(x, y) → –1 as (x, y) → (0, 0) along the y-axis *
(See Figure 4.)

cont’d

*Example 1 – Solution*

*Since f has two different limits along two different lines, the *
given limit does not exist. (This confirms the conjecture we
made on the basis of numerical evidence at the beginning
of this section.)

cont’d

### Limits and Continuity

*Now let’s look at limits that do exist. Just as for functions of *
one variable, the calculation of limits for functions of two
variables can be greatly simplified by the use of properties
of limits.

The Limit Laws can be extended to functions of two

variables: the limit of a sum is the sum of the limits, the limit of a product is the product of the limits, and so on.

In particular, the following equations are true.

### Continuity

### Continuity

*We know that evaluating limits of continuous functions of a *
single variable is easy.

It can be accomplished by direct substitution because the
defining property of a continuous function is
lim_{x→a}*f(x) = f(a).*

Continuous functions of two variables are also defined by the direct substitution property.

### Continuity

*The intuitive meaning of continuity is that if the point (x, y) *
*changes by a small amount, then the value of f(x, y) *

changes by a small amount.

This means that a surface that is the graph of a continuous function has no hole or break.

Using the properties of limits, you can see that sums,

differences, products, and quotients of continuous functions are continuous on their domains.

Let’s use this fact to give examples of continuous functions.

### Continuity

**A polynomial function of two variables (or polynomial, **
*for short) is a sum of terms of the form cx*^{m}*y*^{n}*, where c is a *
*constant and m and n are nonnegative integers.*

**A rational function is a ratio of polynomials.**

For instance,

*f(x, y) = x*^{4} *+ 5x*^{3}*y*^{2} *+ 6xy*^{4} *– 7y + 6*
is a polynomial, whereas

### Continuity

The limits in (2) show that the functions
*f(x, y) = x, g(x, y) = y, and h(x, y) = c are continuous. *

Since any polynomial can be built up out of the simple

*functions f, g, and h by multiplication and addition, it follows *
*that all polynomials are continuous on .*

Likewise, any rational function is continuous on its domain because it is a quotient of continuous functions.

### Example 5

Evaluate

Solution:

*Since f(x, y) = x*^{2}*y*^{3} *– x*^{3}*y*^{2} *+ 3x + 2y is a polynomial, it is *
continuous everywhere, so we can find the limit by direct
substitution:

*(x*^{2}*y*^{3} *– x*^{3}*y*^{2} *+ 3x + 2y) = 1*^{2} 2^{3} – 1^{3} 2^{2} + 3 1
+ 2 2

= 11

### Continuity

Just as for functions of one variable, composition is another way of combining two continuous functions to get a third.

*In fact, it can be shown that if f is a continuous function of *
*two variables and g is a continuous function of a single *
*variable that is defined on the range of f, then the *

*composite function h = g *_{°} *f defined by h(x, y) = g(f(x, y)) is *
also a continuous function.

### Functions of Three or More

### Variables

### Functions of Three or More Variables

Everything that we have done in this section can be extended to functions of three or more variables.

The notation

*means that the values of f(x, y, z) approach the number L*
*as the point (x, y, z) approaches the point (a, b, c) along *
*any path in the domain of f.*

### Functions of Three or More Variables

*Because the distance between two points (x, y, z) and *
*(a, b, c) in is given by , *
we can write the precise definition as follows: for every
number *ε > 0 there is a corresponding number *δ > 0 such
that

*if (x, y, z) is in the domain of f and*

0 < < δ
*then | f(x, y, z) – L | < ε*

### Functions of Three or More Variables

**The function f is continuous at (a, b, c) if**

For instance, the function

is a rational function of three variables and so is continuous
*at every point in except where x*^{2} *+ y*^{2} *+ z*^{2} = 1. In other
words, it is discontinuous on the sphere with center the
origin and radius 1.

### Functions of Three or More Variables

We can write the definitions of a limit for functions of two or three variables in a single compact form as follows.