14.2: LIMITS AND CONTINUITY
KIAM HEONG KWA
1.
Limits of Functions
Let
f
be a function of two variables whose domain
D
includes points
arbitrarily close to (
a,b
). We say that the limit
of
f
(
x,y
) as (
x,y
)
approaches (
a,b
) is the number
L
and we write
•
lim
(
x,y
)
→
(
a,b
)
f
(
x,y
) =
L
,
•
lim
x
→
a,y
→
b
f
(
x,y
) =
L
, or
•
f
(
x,y
)
→
L
as (
x,y
)
→
(
a,b
)
if for every
± >
0 there is a corresponding
δ
=
δ
(
±
)
>
0 such that

f
(
x,y
)

L

< ±
whenever (
x,y
)
∈
D
and
0
<
p
(
x

a
)
2
+ (
y

b
)
2
< δ.
Geometrically, this means that if we restrict (
x,y
) to the interior of the
disk centered at (
a,b
) and of radius
δ
, then the corresponding part of
the graph of
f
lies strictly between the planes
z
=
L

±
and
z
=
L
+
±
.
See ﬁgure 2 on p. 871 of the text for a graphical illustration.
Limit Theorems.
Let
f
and
g
be functions of two variables such that
f
(
x,y
)
→
L
and
g
(
x,y
)
→
M
as (
x,y
)
→
(
a,b
), and let
h
be a function
of one variable. Then
•
lim
(
x,y