Section 15.2
Limits and Continuity
“Generalizing Ideas from Single Variable Calculus”
The ideas of limits and continuity were critical when defining the der
vative in single variable calculus 
f
′
(
x
) = lim
h
→
0
f
(
x
+
h
)
−
f
(
x
)
h
.
We shall generalize these ideas to functions of more than one variable.
1.
Limits of Functions of Two Variables
Recall that the naive idea of the limit of a function
f
(
x
) at a point
x
=
a
is the following: it is the value
f
(
x
) tends towards as
x
gets
close to
a
, where by “close”, we mean

x
−
a

is sufficiently small. This
causes a problem when defining a limit of a function of two variables
 the value

(
x, y
)
−
(
a, b
)

makes no sense. However, the term “close”
refers to distance, and we have a general formula for distance between
points in the plane. Specifically, two points (
x, y
) and (
a, b
) are close
provided
radicalbig
(
x
−
a
)
2
+ (
y
−
b
)
2
is sufficiently small.
We can use this
definition to define the limit at a point of a function of two variables.
Definition 1.1.
Let
f
be a function of two variables whose domain
includes points arbitrarily close to (
a, b
) (though not necessarily (
a, b
)).
Then we say the limit of
f
(
x, y
) as (
x, y
) approaches (
a, b
) is
L
and write
lim
(
x,y
)
→
(
a,b
)
f
(
x, y
) =
L
if for every number
ε >
0, there exists a number
δ >
0 such that

f
(
x, y
)
−
L

< ε
whenever (
x, y
) is in the domain of
f
and
radicalbig
(
x
−
a
)
2
+ (
y
−
b
)
2
< δ.
Visually, since the domain of
f
(
x, y
) is a region in 2space, this means
that if we choose a point (
x, y
) within a ball of radius
δ >
0 centered
at (
a, b
), then the value

f
(
a, b
)
−
f
(
x, y
)

< ε
. The major difference
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 Fall '08
 Stefanov
 Calculus, Continuity, Limits, Limit, lim, Continuous function

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