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In this section, we will learn about: Limits and continuity of various types of functions.
2.
PARTIAL DERIVATIVES
14.2
Limits and Continuity
In this section, we will learn about:
Limits and continuity of
various types of functions.
3.
LIMITS AND CONTINUITY
Let’s compare the behavior of the functions
2 2 2 2
sin( x y ) x y
f ( x, y ) 2 2
and g ( x, y ) 2 2
x y x y
as x and y both approach 0
(and thus the point (x, y) approaches
the origin).
4.
LIMITS AND CONTINUITY
The following tables show values of f(x, y)
and g(x, y), correct to three decimal places,
for points (x, y) near the origin.
5.
LIMITS AND CONTINUITY Table 1
This table shows values of f(x, y).
6.
LIMITS AND CONTINUITY Table 2
This table shows values of g(x, y).
7.
LIMITS AND CONTINUITY
Notice that neither function is defined
at the origin.
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.
8.
LIMITS AND CONTINUITY
It turns out that these guesses based on
numerical evidence are correct.
Thus, we write:
sin( x 2 y 2 )
lim 2 2
1
( x , y ) (0,0) x y
2 2
lim
x y
2 2
does not exist.
( x , y ) (0,0) x y
9.
LIMITS AND CONTINUITY
In general, we use the notation
lim f ( x, y ) L
( x , y ) ( a ,b )
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.
10.
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.
11.
LIMIT OF A FUNCTION Definition 1
Let f be a function of two variables
whose domain D includes points arbitrarily
close to (a, b).
Then, we say that the limit of f(x, y)
as (x, y) approaches (a, b) is L.
12.
LIMIT OF A FUNCTION Definition 1
We write
lim f ( x, y ) L
( x , y ) ( a ,b )
For every number ε > 0, there is a corresponding
number δ > 0 such that,
2 2
if ( x, y ) D and 0 ( x a ) ( y b)
then | f ( x, y ) L |
13.
LIMIT OF A FUNCTION
Other notations for the limit in Definition 1
lim f ( x, y ) L
x a
y b
f ( x, y ) L as ( x, y ) (a, b)
14.
LIMIT OF A FUNCTION
Notice that:
| f ( x, y ) L | is the distance between
the numbers f(x, y) and L
( x a) 2 ( y b) 2 is the distance between
the point (x, y) and the point (a, b).
15.
LIMIT OF A FUNCTION
Thus, 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).
16.
LIMIT OF A FUNCTION
The figure illustrates Definition 1
by means of an arrow diagram.
17.
LIMIT OF A FUNCTION
If any small interval (L – ε, L + ε) is given around L,
then we can find a disk Dδ with center (a, b) and
radius δ > 0 such that:
f maps all the points in Dδ [except possibly (a, b)]
into the interval (L – ε, L + ε).
18.
LIMIT OF A FUNCTION
Another illustration of
Definition 1
is given here, where the
surface S
is the graph of f.
19.
LIMIT OF A FUNCTION
If ε > 0 is given, we can find δ > 0 such that,
if (x, y) is restricted to lie in the disk Dδ and
(x, y) ≠ (a, b), then
The corresponding
part of S lies between
the horizontal planes
z = L – ε and
z = L + ε.
20.
SINGLE VARIABLE FUNCTIONS
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 recall from Chapter 2 that, if lim f ( x) lim f ( x),
x a x a
then lim f ( x) does not exist.
x a
21.
DOUBLE VARIABLE FUNCTIONS
For functions of two
variables, the situation
is not as simple.
22.
DOUBLE VARIABLE FUNCTIONS
This is because we can let (x, y) approach
(a, b) from an infinite number of directions
in any manner whatsoever as long as (x, y)
stays within the domain of f.
23.
LIMIT OF A FUNCTION
Definition 1 refers only to the distance
between (x, y) and (a, b).
It does not refer to the direction of approach.
24.
LIMIT OF A FUNCTION
Therefore, if the limit exists, then f(x, y)
must approach the same limit no matter
how (x, y) approaches (a, b).
25.
LIMIT OF A FUNCTION
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 f ( x, y ) does not exist.
( x , y ) ( a ,b )
26.
LIMIT OF A FUNCTION
f(x, y) → L1 as (x, y) → (a, b) along a path C1
f(x, y) → L2 as (x, y) → (a, b) along a path C2,
where L1 ≠ L2,
then lim f ( x, y )
( x , y ) ( a ,b )
does not exist.
27.
LIMIT OF A FUNCTION Example 1
Show that 2
x y 2
lim
( x , y ) (0,0) x 2 y 2
does not exist.
Let f(x, y) = (x2 – y2)/(x2 + y2).
28.
LIMIT OF A FUNCTION Example 1
First, let’s approach (0, 0) along
the x-axis.
Then, y = 0 gives f(x, 0) = x2/x2 = 1 for all x ≠ 0.
So, f(x, y) → 1 as (x, y) → (0, 0) along the x-axis.
29.
LIMIT OF A FUNCTION Example 1
We now approach along the y-axis by
putting x = 0.
Then, f(0, y) = –y2/y2 = –1 for all y ≠ 0.
So, f(x, y) → –1 as (x, y) → (0, 0) along the y-axis.
30.
LIMIT OF A FUNCTION Example 1
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 the section.
31.
LIMIT OF A FUNCTION Example 2
If xy
f ( x, y ) 2 2
x y
lim f ( x, y )
( x , y ) (0,0)
32.
LIMIT OF A FUNCTION Example 2
If y = 0, then f(x, 0) = 0/x2 = 0.
Therefore,
f(x, y) → 0 as (x, y) → (0, 0) along the x-axis.
33.
LIMIT OF A FUNCTION Example 2
If x = 0, then f(0, y) = 0/y2 = 0.
So,
f(x, y) → 0 as (x, y) → (0, 0) along the y-axis.
34.
LIMIT OF A FUNCTION Example 2
Although we have obtained identical limits
along the axes, that does not show that
the given limit is 0.
35.
LIMIT OF A FUNCTION Example 2
Let’s now approach (0, 0) along another
line, say y = x.
For all x ≠ 0,
x2 1
f ( x, x ) 2 2
x x 2
Therefore,
f ( x, y ) 1
2 as ( x, y ) (0, 0) along y x
36.
LIMIT OF A FUNCTION Example 2
Since we have obtained different limits
along different paths, the given limit does
not exist.
37.
LIMIT OF A FUNCTION
This figure sheds
some light on
Example 2.
The ridge that occurs
above the line y = x
corresponds to the fact
that f(x, y) = ½ for all
points (x, y) on that line
except the origin.
38.
LIMIT OF A FUNCTION Example 3
If xy 2
f ( x, y ) 2 4
x y
lim f ( x, y )
( x , y ) (0,0)
39.
LIMIT OF A FUNCTION Example 3
With the solution of Example 2 in mind,
let’s try to save time by letting (x, y) → (0, 0)
along any nonvertical line through the origin.
40.
LIMIT OF A FUNCTION Example 3
Then, y = mx, where m is the slope,
and f ( x, y ) f ( x, mx)
x(mx) 2
2 4
x (mx)
m2 x3
2 4 4
x m x
m2 x
4 2
1 m x
41.
LIMIT OF A FUNCTION Example 3
f(x, y) → 0 as (x, y) → (0, 0) along y = mx
Thus, f has the same limiting value along
every nonvertical line through the origin.
42.
LIMIT OF A FUNCTION Example 3
However, that does not show that
the given limit is 0.
This is because, if we now let
(x, y) → (0, 0) along the parabola x = y2
we have: 2 2 4
2 y y y 1
f ( x, y ) f ( y , y ) 2 2 4
4
(y ) y 2y 2
So,
f(x, y) → ½ as (x, y) → (0, 0) along x = y2
43.
LIMIT OF A FUNCTION Example 3
Since different paths lead to different
limiting values, the given limit does not
44.
LIMIT OF A FUNCTION
Now, let’s look at limits
that do exist.
45.
LIMIT OF A FUNCTION
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.
46.
LIMIT OF A FUNCTION
The Limit Laws listed in Section 2.3 can be
extended to functions of two variables.
For instance,
The limit of a sum is the sum of the limits.
The limit of a product is the product of the limits.
47.
LIMIT OF A FUNCTION Equations 2
In particular, the following equations
are true.
lim x a
( x , y ) ( a ,b )
lim y b
( x , y ) ( a ,b )
lim c c
( x , y ) ( a ,b )
48.
LIMIT OF A FUNCTION Equations 2
The Squeeze Theorem
also holds.
49.
LIMIT OF A FUNCTION Example 4
Find 2
3x y
lim
( x , y ) (0,0) x 2 y 2
if it exists.
50.
LIMIT OF A FUNCTION Example 4
As in Example 3, we could show that
the limit along any line through the origin
is 0.
However, this doesn’t prove that
the given limit is 0.
51.
LIMIT OF A FUNCTION Example 4
However, the limits along the parabolas
y = x2 and x = y2 also turn out to be 0.
So, we begin to suspect that the limit
does exist and is equal to 0.
52.
LIMIT OF A FUNCTION Example 4
Let ε > 0.
We want to find δ > 0 such that
2
2 2 3x y
if 0 x y then 2 2
0
x y
2
2 2 3x | y |
that is, if 0 x y then 2 2
x y
53.
LIMIT OF A FUNCTION Example 4
x2 ≤ x2 = y2 since y2 ≥ 0
Thus,
x2/(x2 + y2) ≤ 1
54.
LIMIT OF A FUNCTION E. g. 4—Equation 3
2
3x | y | 2 2 2
2 2
3 | y | 3 y 3 x y
x y
55.
LIMIT OF A FUNCTION Example 4
Thus, if we choose δ = ε/3
2 2
and let 0 x y
3x 2 y 2 2
2 2
0 3 x y 3 3
x y 3
56.
LIMIT OF A FUNCTION Example 4
Hence, by Definition 1,
2
3x y
lim 0
( x , y ) (0,0) x 2 y 2
57.
CONTINUITY OF SINGLE VARIABLE FUNCTIONS
Recall that evaluating limits of continuous
functions of a single variable is easy.
It can be accomplished by direct substitution.
This is because the defining property of
a continuous function is lim f ( x) f ( a )
x a
58.
CONTINUITY OF DOUBLE VARIABLE FUNCTIONS
Continuous functions of two variables
are also defined by the direct substitution
59.
CONTINUITY Definition 4
A function f of two variables is called
continuous at (a, b) if
lim f ( x, y ) f ( a , b )
( x , y ) ( a ,b )
We say f is continuous on D if f is
continuous at every point (a, b) in D.
60.
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
This means that a surface that is the graph of
a continuous function has no hole or break.
61.
Using the properties of limits, you can see
that sums, differences, products, quotients
of continuous functions are continuous on
their domains.
Let’s use this fact to give examples
of continuous functions.
62.
A polynomial function of two variables
(polynomial, for short) is a sum of terms
of the form cxmyn,
c is a constant.
m and n are nonnegative integers.
63.
RATIONAL FUNCTION
A rational function is
a ratio of polynomials.
64.
RATIONAL FUNCTION VS. POLYNOMIAL
4 3 2 4
f ( x, y ) x 5 x y 6 xy 7 y 6
is a polynomial.
2 xy 1
g ( x, y ) 2 2
x y
is a rational function.
65.
The limits in Equations 2 show that
the functions
f(x, y) = x, g(x, y) = y, h(x, y) = c
are continuous.
66.
CONTINUOUS POLYNOMIALS
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 R2.
67.
CONTINUOUS RATIONAL FUNCTIONS
Likewise, any rational function is
continuous on its domain because it is
a quotient of continuous functions.
68.
CONTINUITY Example 5
2 3 3 2
lim ( x y x y 3 x 2 y )
( x , y ) (1,2)
2 3 3 2
f ( x, y ) x y x y 3x 2 y is a polynomial.
Thus, it is continuous everywhere.
69.
CONTINUITY Example 5
Hence, we can find the limit by direct
substitution:
2 3 3 2
lim ( x y x y 3x 2 y )
( x , y ) (1,2)
2 3 3 2
1 2 1 2 3 1 2 2
11
70.
CONTINUITY Example 6
Where is the function
2 2
x y
f ( x, y ) 2 2
x y
71.
CONTINUITY Example 6
The function f is discontinuous at (0, 0)
because it is not defined there.
Since f is a rational function, it is continuous
on its domain, which is the set
D = {(x, y) | (x, y) ≠ (0, 0)}
72.
CONTINUITY Example 7
2 2
Let x y
2 2
if ( x, y ) (0, 0)
g ( x, y ) x y
0 if ( x, y ) (0, 0)
Here, g is defined at (0, 0).
However, it is still discontinuous there because
lim g ( x, y )
( x , y ) (0,0)
does not exist (see Example 1).
73.
CONTINUITY Example 8
2
3x y
2 2
if ( x, y ) (0, 0)
f ( x, y ) x y
0 if ( x, y ) (0, 0)
74.
CONTINUITY Example 8
We know f is continuous for (x, y) ≠ (0, 0)
since it is equal to a rational function there.
Also, from Example 4, we have:
2
3x y
lim f ( x, y ) lim
( x , y ) (0,0) ( x , y ) (0,0) x 2 y 2
0 f (0, 0)
75.
CONTINUITY Example 8
Thus, f is continuous at (0, 0).
So, it is continuous on R2.
76.
This figure shows the
graph of
the continuous function
in Example 8.
77.
COMPOSITE FUNCTIONS
Just as for functions of one variable,
composition is another way of combining
two continuous functions to get a third.
78.
COMPOSITE FUNCTIONS
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
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.
79.
COMPOSITE FUNCTIONS Example 9
Where is the function h(x, y) = arctan(y/x)
The function f(x, y) = y/x is a rational function
and therefore continuous except on the line x = 0.
The function g(t) = arctan t is continuous
everywhere.
80.
COMPOSITE FUNCTIONS Example 9
So, the composite function
g(f(x, y)) = arctan(y, x) = h(x, y)
is continuous except where x = 0.
81.
COMPOSITE FUNCTIONS Example 9
The figure shows the
break in the graph
of h above the y-axis.
82.
FUNCTIONS OF THREE OR MORE VARIABLES
Everything that we have done in
this section can be extended to functions
of three or more variables.
83.
MULTIPLE VARIABLE FUNCTIONS
The notation
lim f ( x, y , z ) L
( x , y , z ) ( a ,b , c )
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.
84.
MULTIPLE VARIABLE FUNCTIONS
The distance between two points (x, y, z)
and (a, b, c) in R3 is given by:
2 2 2
( x a ) ( y b) ( z c )
Thus, we can write the precise definition
as follows.
85.
MULTIPLE VARIABLE FUNCTIONS
For every number ε > 0, there is
a corresponding number δ > 0 such that,
if (x, y, z) is in the domain of f
2 2 2
and 0 ( x a) ( y b) ( z c)
|f(x, y, z) – L| < ε
86.
MULTIPLE VARIABLE FUNCTIONS
The function f is continuous at (a, b, c)
lim f ( x, y, z ) f (a, b, c)
( x , y , z ) ( a ,b , c )
87.
MULTIPLE VARIABLE FUNCTIONS
For instance, the function
1
f ( x, y , z ) 2 2 2
x y z 1
is a rational function of three variables.
So, it is continuous at every point in R3
except where x2 + y2 + z2 = 1.
88.
MULTIPLE VARIABLE FUNCTIONS
In other words, it is discontinuous
on the sphere with center the origin
and radius 1.
89.
MULTIPLE VARIABLE FUNCTIONS
If we use the vector notation introduced at
the end of Section 10.1, then we can write
the definitions of a limit for functions of two or
three variables in a single compact form as
90.
MULTIPLE VARIABLE FUNCTIONS Equation 5
If f is defined on a subset D of Rn,
then lim f (x) L means that, for every
x a
number ε > 0, there is a corresponding
number δ > 0 such that
if x D and 0 | x a |
then | f ( x) L |
91.
MULTIPLE VARIABLE FUNCTIONS
If n = 1, then
x=x and a=a
So, Equation 5 is just the definition
of a limit for functions of a single variable.
92.
MULTIPLE VARIABLE FUNCTIONS
If n = 2, we have
x =
a =
2 2
| x a | ( x a ) ( y b)
So, Equation 5 becomes Definition 1.
93.
MULTIPLE VARIABLE FUNCTIONS
If n = 3, then
x = and a =
So, Equation 5 becomes the definition of
a limit of a function of three variables.
94.
MULTIPLE VARIABLE FUNCTIONS
In each case, the definition of continuity
can be written as:
lim f (x) f (a)
x a
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