15.2
Limits and Continuity
Recall:
A realvalued function
f
of one variable has a limit
L
as
x
approaches the number
a
, i.e.,
lim
x
→
a
f
(
x
) =
L
, if there is an open interval containing
a
where
f
is defined (except possibly at
x
=
a
),
and if for every positive number
there is a positive number
δ
so that whenever
x
is a number with
0
<

x

a

<
δ
then

f
(
x
)

L

<
.
NOTE: The only way in which to approach the value
a
was along the
x
axis from the left or from
the right. So if
lim
x
→
a

f
(
x
) =
lim
x
→
a
+
f
(
x
) =
L
, then the twosided limit existed and was equal to
L
. On
the other hand, if the two onesided limits didn’t equal, we were able to conclude the twosided limit
didn’t exist.
How do these ideas change when
f
is a realvalued function of two or more variables?
Definition 15.2.1
Let
(
a, b
)
be a point in the
xy
plane and let
r >
0
.
The open disk centered at
(
a, b
)
with radius
r
is denoted
D
r
(
a, b
)
and is given by
D
r
(
a, b
) =
{
(
x, y
)

(
x

a
)
2
+ (
y

b
)
2
< r
}
.
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 Fall '07
 Chung
 Continuity, Multivariable Calculus, Limits, Limit, lim, Limit of a function, x,y

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