The document discusses multi-variable functions and how to visualize them. It defines Rn as the set of n-tuples of real numbers. A n-variable real valued function maps from Rn to the real numbers. Level curves are used to visualize functions of two variables by setting z equal to constants and plotting the resulting equations in the x-y plane. This produces families of curves that depict the shape of the surface. Examples show how different functions produce different level curve shapes like ellipses, hyperbolas or parabolas.

1. Multi-variable Functions

2. Multi-variable Functions
We write R2 = {(x, y)| x and y are real numbers},

3. Multi-variable Functions
We write R2 = {(x, y)| x and y are real numbers},
R3 = {all points (x, y, z)} and in general

4. Multi-variable Functions
We write R2 = {(x, y)| x and y are real numbers},
R3 = {all points (x, y, z)} and in general
Rn = {(x1, x2,.., xn) where xi is a real number for
i = 1, 2,.., n}

5. Multi-variable Functions
We write R2 = {(x, y)| x and y are real numbers},
R3 = {all points (x, y, z)} and in general
Rn = {(x1, x2,.., xn) where xi is a real number for
i = 1, 2,.., n}
A two-variable real valued function is a function f from
R2 to R.
We write it as f(x, y) = z where z is the output.

6. Multi-variable Functions
We write R2 = {(x, y)| x and y are real numbers},
R3 = {all points (x, y, z)} and in general
Rn = {(x1, x2,.., xn) where xi is a real number for
i = 1, 2,.., n}
A two-variable real valued function is a function f from
R2 to R.
We write it as f(x, y) = z where z is the output.
A three-variable real valued function is a function
f from R3 to R.
We write it as f(x, y, z) = w where w is the output.

7. Multi-variable Functions
We write R2 = {(x, y)| x and y are real numbers},
R3 = {all points (x, y, z)} and in general
Rn = {(x1, x2,.., xn) where xi is a real number for
i = 1, 2,.., n}
A two-variable real valued function is a function f from
R2 to R.
We write it as f(x, y) = z where z is the output.
A three-variable real valued function is a function
f from R3 to R.
We write it as f(x, y, z) = w where w is the output.
A n-variable real valued function is a function f from
Rn to R.
We write it as f(x1,x2,..,xn) = y where y is the output.

8. Multi-variable Functions
Example: A. Let f(x, y) = x2 – y2, find f(3, -2) and the
domain of f.

9. Multi-variable Functions
Example: A. Let f(x, y) = x2 – y2, find f(3, -2) and the
domain of f.
Set x = 3, y = -2, then f(3, -2) = 32 – (-2)2

10. Multi-variable Functions
Example: A. Let f(x, y) = x2 – y2, find f(3, -2) and the
domain of f.
Set x = 3, y = -2, then f(3, -2) = 32 – (-2)2 = 9 – 4 = 5.

11. Multi-variable Functions
Example: A. Let f(x, y) = x2 – y2, find f(3, -2) and the
domain of f.
Set x = 3, y = -2, then f(3, -2) = 32 – (-2)2 = 9 – 4 = 5.
The domain is all of R2.

12. Multi-variable Functions
Example: A. Let f(x, y) = x2 – y2, find f(3, -2) and the
domain of f.
Set x = 3, y = -2, then f(3, -2) = 32 – (-2)2 = 9 – 4 = 5.
The domain is all of R2.
x2 + y2
B. Let g(x, y) = 2 2 , find g(0, 2) and the domain of g.
x –y

13. Multi-variable Functions
Example: A. Let f(x, y) = x2 – y2, find f(3, -2) and the
domain of f.
Set x = 3, y = -2, then f(3, -2) = 32 – (-2)2 = 9 – 4 = 5.
The domain is all of R2.
x2 + y2
B. Let g(x, y) = 2 2 , find g(0, 2) and the domain of g.
x –y
22
Set x = 0, y = 2, then g(0, 2) = 2 = -1
–2

14. Multi-variable Functions
Example: A. Let f(x, y) = x2 – y2, find f(3, -2) and the
domain of f.
Set x = 3, y = -2, then f(3, -2) = 32 – (-2)2 = 9 – 4 = 5.
The domain is all of R2.
x2 + y2
B. Let g(x, y) = 2 2 , find g(0, 2) and the domain of g.
x –y
22
Set x = 0, y = 2, then g(0, 2) = 2 = -1
–2
The domain consists of
all points (x, y) in R2 except
x2 – y2 = 0,

15. Multi-variable Functions
Example: A. Let f(x, y) = x2 – y2, find f(3, -2) and the
domain of f.
Set x = 3, y = -2, then f(3, -2) = 32 – (-2)2 = 9 – 4 = 5.
The domain is all of R2.
x2 + y2
B. Let g(x, y) = 2 2 , find g(0, 2) and the domain of g.
x –y
22
Set x = 0, y = 2, then g(0, 2) = 2 = -1
–2
The domain consists of
all points (x, y) in R2 except
x2 – y2 = 0, or (x + y)(x – y) = 0
i.e. y = -x, or y = x

16. Multi-variable Functions
Example: A. Let f(x, y) = x2 – y2, find f(3, -2) and the
domain of f.
Set x = 3, y = -2, then f(3, -2) = 32 – (-2)2 = 9 – 4 = 5.
The domain is all of R2.
x2 + y2
B. Let g(x, y) = 2 2 , find g(0, 2) and the domain of g.
x –y
22
Set x = 0, y = 2, then g(0, 2) = 2 = -1
–2 y=x
The domain consists of
all points (x, y) in R2 except
x2 – y2 = 0, or (x + y)(x – y) = 0
i.e. y = -x, or y = x y=-x

17. Multi-variable Functions
Usually the graph of z = f(x, y) is a surface in 3D
space. We may visualize the surface by the level
curves of the surface.

18. Multi-variable Functions
Usually the graph of z = f(x, y) is a surface in 3D
space. We may visualize the surface by the level
curves of the surface.
The level curves are the same as "contour lines" on a
topographical map. For example:

19. Multi-variable Functions
Usually the graph of z = f(x, y) is a surface in 3D
space. We may visualize the surface by the level
curves of the surface.
The level curves are the same as "contour lines" on a
topographical map. For example:
y z
z=4 z=4
y z=2
z=2
x x
z=0  z=0
z=-2
z=-2
z=-4
z=-4

20. Multi-variable Functions
For example, the contour lines of a circular bowl:
z=0
z=4
z=8
z=10

21. Multi-variable Functions
For example, the contour lines of a circular bowl:
z=10
z=0 z=8
z=4

z=4
z=8
z=10 z=0

22. Multi-variable Functions
For example, the contour lines of a circular bowl:
z=10
z=0 z=8
z=4

z=4
z=8
z=10 z=0
For example, the contour lines of a circular dome:
z=10
z=8
z=4
z=0

23. Multi-variable Functions
For example, the contour lines of a circular bowl:
z=10
z=0 z=8
z=4

z=4
z=8
z=10 z=0
For example, the contour lines of a circular dome:
z=10
z=8
z=10 z=4
z=8

z=0
z=4
z=0 y
x

24. Multi-variable Functions
To obtain level curves, set z to different constants c and
plot the 2D graphs of the resulting equations in x&y in
the planes z = c.

25. Multi-variable Functions
To obtain level curves, set z to different constants c and
plot the 2D graphs of the resulting equations in x&y in
the planes z = c.
x2 y2
Example: z = f(x, y) = +
4 9

26. Multi-variable Functions
To obtain level curves, set z to different constants c and
plot the 2D graphs of the resulting equations in x&y in
the planes z = c.
x2 y2
Example: z = f(x, y) = +
4 9
Set z as constants.

27. Multi-variable Functions
To obtain level curves, set z to different constants c and
plot the 2D graphs of the resulting equations in x&y in
the planes z = c.
x2 y2
Example: z = f(x, y) = +
4 9
Set z as constants.
When z is negative, there is no graph.

28. Multi-variable Functions
To obtain level curves, set z to different constants c and
plot the 2D graphs of the resulting equations in x&y in
the planes z = c.
x2 y2
Example: z = f(x, y) = +
4 9
Set z as constants.
When z is negative, there is no graph.
If z = 0, the ellipses collapses to a point.
If z>0, we get ellipses:

29. Multi-variable Functions
To obtain level curves, set z to different constants c and
plot the 2D graphs of the resulting equations in x&y in
the planes z = c.
x2 y2
Example: z = f(x, y) = +
4 9
Set z as constants.
When z is negative, there is no graph.
If z = 0, the ellipses collapses to a point.
If z>0, we get ellipses:
x2 + y2
z=1: 1 = , with axes 2 and 3.
4 9

30. Multi-variable Functions
To obtain level curves, set z to different constants c and
plot the 2D graphs of the resulting equations in x&y in
the planes z = c.
x2 y2
Example: z = f(x, y) = + z
4 9
Set z as constants.
When z is negative, there is no graph.
If z = 0, the ellipses collapses to a point.
If z>0, we get ellipses: 2 3
y
z=1: 1 = x + y
2 2
, with axes 2 and 3.
4 9
x

31. Multi-variable Functions
To obtain level curves, set z to different constants c and
plot the 2D graphs of the resulting equations in x&y in
the planes z = c.
x2 y2
Example: z = f(x, y) = + z
4 9
Set z as constants.
When z is negative, there is no graph.
6
If z = 0, the ellipses collapses to a point. 4
If z>0, we get ellipses: 2 3
y
z=1: 1 = x + y
2 2
, with axes 2 and 3.
4 9
x + y2
2
z=4: 1 = 16 36
, with axes 4 and 6.
x

32. Multi-variable Functions
To obtain level curves, set z to different constants c and
plot the 2D graphs of the resulting equations in x&y in
the planes z = c.
x2 y2
Example: z = f(x, y) = + z
4 9
Set z as constants.
When z is negative, there is no graph.
6
If z = 0, the ellipses collapses to a point. 4
If z>0, we get ellipses: 2 3
y
z=1: 1 = x + y
2 2
, with axes 2 and 3.
4 9
x + y2
2
z=4: 1 = 16 36
, with axes 4 and 6.
x
The greater the z is the larger the ellipse is.

33. Multi-variable Functions
To obtain level curves, set z to different constants c and
plot the 2D graphs of the resulting equations in x&y in
the planes z = c.
x2 y2
Example: z = f(x, y) = + z
4 9
Set z as constants.
When z is negative, there is no graph.
6
If z = 0, the ellipses collapses to a point. 4
If z>0, we get ellipses: 2 3
y
z=1: 1 = x + y
2 2
, with axes 2 and 3.
4 9
x + y2
2
z=4: 1 = 16 36
, with axes 4 and 6.
x
The greater the z is the larger the ellipse is.
Set x = 0, get parabola as the outline of the stacked ellipses.

34. Multi-variable Functions
We get the elliptical parabloid.
y
x

35. Multi-variable Functions
Example: z = x2 – y2

36. Multi-variable Functions
Example: z = x2 – y2
Set z = 1, 4, we get the
equations: x2 – y2 = 1,
x2 – y2 = 4
which are hyperbolas
open in the x direction.

37. Multi-variable Functions
Example: z = x2 – y2
Set z = 1, 4, we get the y
equations: x2 – y2 = 1,
x2 – y2 = 4
which are hyperbolas
open in the x direction. z=1 z=1
-2 -1 1 2 x

38. Multi-variable Functions
Example: z = x2 – y2
Set z = 1, 4, we get the y
equations: x2 – y2 = 1,
x2 – y2 = 4
which are hyperbolas
open in the x direction. z=4
z=1 z=1
z=4
-2 -1 1 2 x

39. Multi-variable Functions
Example: z = x2 – y2
Set z = 1, 4, we get the y
equations: x2 – y2 = 1,
x2 – y2 = 4
which are hyperbolas
open in the x direction. z=4
z=1 z=1
z=4
-2 -1 1 2 x
Set z = -1, -4, we get the z=-1
equations: y2 – x2 =1,
z=-4
y2 – x2 =4
which are hyperbolas
open in the y direction.

40. Multi-variable Functions
z
y
x
z = x2 – y2
We get the saddle with the origin as a saddle point.

41. Multi-variable Functions
1
Example: Let z = , construct some contour
x +y
2 2
lines then sketch its graph.

42. Multi-variable Functions
1
Example: Let z = , construct some contour
x +y
2 2
lines then sketch its graph.
z has to be positive, and x2 + y2 = 1 .
z

43. Multi-variable Functions
1
Example: Let z = , construct some contour
x +y
2 2
lines then sketch its graph.
z has to be positive, and x2 + y2 = 1 .
z
Select z=1/9, ¼, 1, 4, 9, we get -

44. Multi-variable Functions
1
Example: Let z = , construct some contour
x +y
2 2
lines then sketch its graph.
z has to be positive, and x2 + y2 = 1 .
z
Select z=1/9, ¼, 1, 4, 9, we get -
z =1/9: x2 + y2 = 9
z =1/4: x2 + y2 = 4
z =1: x2 + y2 = 1
z =4: x2 + y2 = 1/4
z =9: x2 + y2 = 1/9

45. Multi-variable Functions
1
Example: Let z = , construct some contour
x +y
2 2
lines then sketch its graph.
z has to be positive, and x2 + y2 = 1 .
z
Select z=1/9, ¼, 1, 4, 9, we get -
z =1/9: x2 + y2 = 9
z =1/4: x2 + y2 = 4
z =1: x2 + y2 = 1
z =4: x2 + y2 = 1/4
z =9: x2 + y2 = 1/9

46. Multi-variable Functions
We get the following surface:
1
Z= 2
x + y2

47. Multi-variable Functions
1
Example: Let z = , construct some contour
x –y
2 2
lines then sketch its graph.

48. Multi-variable Functions
1
Example: Let z = , construct some contour
x –y
2 2
lines then sketch its graph.
The domain is x2 – y2 = 0 i.e (x – y)(x + y) = 0

49. Multi-variable Functions
1
Example: Let z = , construct some contour
x –y
2 2
lines then sketch its graph.
The domain is x2 – y2 = 0 i.e (x – y)(x + y) = 0
y
x

50. Multi-variable Functions
1
Example: Let z = , construct some contour
x –y
2 2
lines then sketch its graph.
The domain is x2 – y2 = 0 i.e (x – y)(x + y) = 0
z may be any number except 0.

51. Multi-variable Functions
1
Example: Let z = , construct some contour
x –y
2 2
lines then sketch its graph.
The domain is x2 – y2 = 0 i.e (x – y)(x + y) = 0
z may be any number except 0.
y
Select z= -1, -¼, ¼ , 1
z =1: x2 – y2 = 1
z =1/4 : x2 – y2 = 4
x

52. Multi-variable Functions
1
Example: Let z = , construct some contour
x –y
2 2
lines then sketch its graph.
The domain is x2 – y2 = 0 i.e (x – y)(x + y) = 0
z may be any number except 0.
y
Select z= -1, -¼, ¼ , 1
z =1: x2 – y2 = 1
z =1/4 : x2 – y2 = 4 z=1/4 z=1/4
z=1 z=1
-2 -1 1 2 x

53. Multi-variable Functions
1
Example: Let z = , construct some contour
x –y
2 2
lines then sketch its graph.
The domain is x2 – y2 = 0 i.e (x – y)(x + y) = 0
z may be any number except 0.
y
Select z= -1, -¼, ¼ , 1
z =1: x2 – y2 = 1
z =1/4 : x2 – y2 = 4 z=1/4 z=1/4
z=1 z=1
z =-1/4 : y2 – x2 = 4 -2 -1 1 2 x
z =-1 : y2 – x2 = 1

54. Multi-variable Functions
1
Example: Let z = , construct some contour
x –y
2 2
lines then sketch its graph.
The domain is x2 – y2 = 0 i.e (x – y)(x + y) = 0
z may be any number except 0.
y
Select z= -1, -¼, ¼ , 1
z =1: x2 – y2 = 1
z =1/4 : x2 – y2 = 4 z=1/4 z=1/4
z=1 z=1
z =-1/4 : y2 – x2 = 4 -2 -1 1 2 x
z=-1
z =-1 : y2 – x2 = 1 z=-1/4

55. Multi-variable Functions
We get the following surface:
z
y
x
1
Z= 2
x – y2

56. Multi-variable Functions
For a function of three variables w = f(x, y, z),
we may set w to constants to get the level surfaces.

57. Multi-variable Functions
For a function of three variables w = f(x, y, z),
we may set w to constants to get the level surfaces.
Example: Let w = f(x, y, z) = x2 + y2 + z2,
sketch the level surfaces for w = 1, 4, and 9

58. Multi-variable Functions
For a function of three variables w = f(x, y, z),
we may set w to constants to get the level surfaces.
Example: Let w = f(x, y, z) = x2 + y2 + z2,
sketch the level surfaces for w = 1, 4, and 9
These surfaces are spheres:

59. Multi-variable Functions
For a function of three variables w = f(x, y, z),
we may set w to constants to get the level surfaces.
Example: Let w = f(x, y, z) = x2 + y2 + z2,
sketch the level surfaces for w = 1, 4, and 9
These surfaces are spheres:

60. Multi-variable Functions
For a function of three variables w = f(x, y, z),
we may set w to constants to get the level surfaces.
Example: Let w = f(x, y, z) = x2 + y2 + z2,
sketch the level surfaces for w = 1, 4, and 9
These surfaces are spheres:

61. Multi-variable Functions
For a function of three variables w = f(x, y, z),
we may set w to constants to get the level surfaces.
Example: Let w = f(x, y, z) = x2 + y2 + z2,
sketch the level surfaces for w = 1, 4, and 9
These surfaces are spheres:

62. Limits of Multi-variable Functions
For real valued function f(x), we could have the right
hand limit (as xa+) different from and left hand limit
(as xa-) hence the limit fails to exist, as xa .

63. Limits of Multi-variable Functions
For real valued function f(x), we could have the right
hand limit (as xa+) different from and left hand limit
(as xa-) hence the limit fails to exist, as xa .
Example:
-1, for x < 0
f(x) =
1, for x > 0

64. Limits of Multi-variable Functions
For real valued function f(x), we could have the right
hand limit (as xa+) different from and left hand limit
(as xa-) hence the limit fails to exist, as xa .
Example:
-1, for x < 0
f(x) =
1, for x > 0

65. Limits of Multi-variable Functions
For real valued function f(x), we could have the right
hand limit (as xa+) different from and left hand limit
(as xa-) hence the limit fails to exist, as xa .
Example:
-1, for x < 0
f(x) =
1, for x > 0
lim f(x) = -1, lim f(x) = 1,
x0- x0+

66. Limits of Multi-variable Functions
For real valued function f(x), we could have the right
hand limit (as xa+) different from and left hand limit
(as xa-) hence the limit fails to exist, as xa .
Example:
-1, for x < 0
f(x) =
1, for x > 0
lim f(x) = -1, lim f(x) = 1,
x0- x0+
Therefore lim f(x) doesn't exist as x0.

67. Limits of Multi-variable Functions
For real valued function f(x), we could have the right
hand limit (as xa+) different from and left hand limit
(as xa-) hence the limit fails to exist, as xa .
Example:
-1, for x < 0
f(x) =
1, for x > 0
lim f(x) = -1, lim f(x) = 1,
x0- x0+
Therefore lim f(x) doesn't exist as x0.
In fact, the limit exists if and only if the right and left
hand limits for function of one variable f(x).

68. Limits of Multi-variable Functions
In 2D, there are infinitely many "sides" to a point (a, b).
To clarify the meaning of "(x, y) approches (a, b)", we specify
the path (x, y) takes.

69. Limits of Multi-variable Functions
In 2D, there are infinitely many "sides" to a point (a, b).
To clarify the meaning of "(x, y) approches (a, b)", we specify
the path (x, y) takes.
Let C be a path containing (a, b), if (x, y) approches (a, b) while
staying on the path C, then we say "(x, y) approaches (a, b)
along C" and its written as "(x, y)(a, b) along C".

70. Limits of Multi-variable Functions
In 2D, there are infinitely many "sides" to a point (a, b).
To clarify the meaning of "(x, y) approches (a, b)", we specify
the path (x, y) takes.
Let C be a path containing (a, b), if (x, y) approches (a, b) while
staying on the path C, then we say "(x, y) approaches (a, b)
along C" and its written as "(x, y)(a, b) along C". C
(x, y)
(a, b)
(x, y)  (a, b)
along C.

71. Limits of Multi-variable Functions
In 2D, there are infinitely many "sides" to a point (a, b).
To clarify the meaning of "(x, y) approches (a, b)", we specify
the path (x, y) takes.
Let C be a path containing (a, b), if (x, y) approches (a, b) while
staying on the path C, then we say "(x, y) approaches (a, b)
along C" and its written as "(x, y)(a, b) along C". C
We use this notation to talk about limits of f(x, y).
(x, y)
(a, b)
(x, y)  (a, b)
along C.

72. Limits of Multi-variable Functions
In 2D, there are infinitely many "sides" to a point (a, b).
To clarify the meaning of "(x, y) approches (a, b)", we specify
the path (x, y) takes.
Let C be a path containing (a, b), if (x, y) approches (a, b) while
staying on the path C, then we say "(x, y) approaches (a, b)
along C" and its written as "(x, y)(a, b) along C". C
We use this notation to talk about limits of f(x, y).
Example: A. Let C be the path y = 2x (x, y)
(a, b)
and (0,0) be a point on C. Let f(x, y) = y/x. (x, y)  (a, b)
Find lim f(x, y) as (x, y)  (0, 0) along C. along C.

73. Limits of Multi-variable Functions
In 2D, there are infinitely many "sides" to a point (a, b).
To clarify the meaning of "(x, y) approches (a, b)", we specify
the path (x, y) takes.
Let C be a path containing (a, b), if (x, y) approches (a, b) while
staying on the path C, then we say "(x, y) approaches (a, b)
along C" and its written as "(x, y)(a, b) along C". C
We use this notation to talk about limits of f(x, y).
Example: A. Let C be the path y = 2x (x, y)
(a, b)
and (0,0) be a point on C. Let f(x, y) = y/x. (x, y)  (a, b)
Find lim f(x, y) as (x, y)  (0, 0) along C. along C.
f(x, y) = 2x/x = 2 for (x, y) along the curve y = 2x.

74. Limits of Multi-variable Functions
In 2D, there are infinitely many "sides" to a point (a, b).
To clarify the meaning of "(x, y) approches (a, b)", we specify
the path (x, y) takes.
Let C be a path containing (a, b), if (x, y) approches (a, b) while
staying on the path C, then we say "(x, y) approaches (a, b)
along C" and its written as "(x, y)(a, b) along C". C
We use this notation to talk about limits of f(x, y).
Example: A. Let C be the path y = 2x (x, y)
(a, b)
and (0,0) be a point on C. Let f(x, y) = y/x. (x, y)  (a, b)
Find lim f(x, y) as (x, y)  (0, 0) along C. along C.
f(x, y) = 2x/x = 2 for (x, y) along the curve y = 2x.
Hence lim f(x, y) = lim 2 = 2 as (x, y)  (0, 0) along C.
Note that if C is y = 3x then the answer is 3.

75. Limits of Multi-variable Functions
Definition of Limits for f(x, y):
We say lim f(x,y) = L as (x, y)(a, b)
if and only if that
for all continuous curves C, we get lim f(x,y) = L
as (x, y)(a, b) along C,.

76. Limits of Multi-variable Functions
Definition of Limits for f(x, y):
We say lim f(x,y) = L as (x, y)(a, b)
if and only if that
for all continuous curves C, we get lim f(x,y) = L
as (x, y)(a, b) along C,.
Theorem: If f(x, y) is an elementary function and
(a, b) is in the domain of f(x, y), then
lim f(x, y) = f(a, b) as (x, y)  (a, b).

77. Limits of Multi-variable Functions
Definition of Limits for f(x, y):
We say lim f(x,y) = L as (x, y)(a, b)
if and only if that
for all continuous curves C, we get lim f(x,y) = L
as (x, y)(a, b) along C,.
Theorem: If f(x, y) is an elementary function and
(a, b) is in the domain of f(x, y), then
lim f(x, y) = f(a, b) as (x, y)  (a, b).
x2+y2
Example: lim e =1
(x, y)(0,0)

78. Limits of Multi-variable Functions
Definition of Limits for f(x, y):
We say lim f(x,y) = L as (x, y)(a, b)
if and only if that
for all continuous curves C, we get lim f(x,y) = L
as (x, y)(a, b) along C,.
Theorem: If f(x, y) is an elementary function and
(a, b) is in the domain of f(x, y), then
lim f(x, y) = f(a, b) as (x, y)  (a, b).
x2+y2
Example: lim e =1
(x, y)(0,0)
x4 – y4
Example: lim
(x, y)(0,0) x2+y2

79. Limits of Multi-variable Functions
Definition of Limits for f(x, y):
We say lim f(x,y) = L as (x, y)(a, b)
if and only if that
for all continuous curves C, we get lim f(x,y) = L
as (x, y)(a, b) along C,.
Theorem: If f(x, y) is an elementary function and
(a, b) is in the domain of f(x, y), then
lim f(x, y) = f(a, b) as (x, y)  (a, b).
x2+y2
Example: lim e =1
(x, y)(0,0)
x4 – y4
Example: lim = lim x2 – y2 = 0
(x, y)(0,0) x +y
2 2
(x, y)(0,0)