Multivariable Epsilon-Delta Limit Definitions

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The definition of a limit:The expression
lim
xa
f(x)=L
is an abbreviation for: the value of the single-variable function
f(x)
approaches
L
​
as
x
approaches the value
a
. More formally, this means that
f(x)
can be made arbitrarily close to
L
by making
x
sufficiently close to
a
, or in precise mathematical terms, for each real
ϵ>0
, there exists a
δ>0
such that
0<|x-a|<δ|f(x)-L|<ϵ
. In other words, the inequalities state that for all
x
except
a
within
δ
of
a
,
f(x)
is within
ϵ
of
L
.
This definition extends to multivariable functions as distances are measured with the Euclidean metric.
In the figure, the horizontal planes
10±ϵ
represent the bounds on
f(x,y)
and the cylinder is
|x-a|=δ
. No matter what
ϵ
is given, a
δ
is found (represented by the changing radius of the cylinder) so that all points on the surface
z=f(x,y)
inside the cylinder are between the two planes.

Details

For the limit of a multivariable function, consider the two-variable function
f(x,y)
. (Note that the following extends to functions of more than just two variables, but for the sake of simplicity, two-variable functions are discussed.) The same limit definition applies here as in the one-variable case, but because the domain of the function is now defined by two variables, distance is measured as
2
(x-
a
x
)
+
2
(y-
a
y
)
, all pairs
(x,y)
within
δ
of
(
a
x
,
a
y
)
are considered, and
f(x,y)
should be within
ϵ
of
L
for all such pairs
(x,y)
. As an example, here is a proof that the limit of
3
x
y-
3
y
x
390
+10
is 10 as
(x,y)(0,0)
. Claim: for a given
ϵ>0
, choosing
δ=min1,
ϵ
2
satisfies the appropriate conditions for the definition of a limit:
2
(x-
a
x
)
+
2
(y-
a
y
)
<δ
(the given condition) reduces to
2
x
+
2
y
<δ
, which implies that
|x|<δ
and
|y|<δ
.
Now,
|f(x,y)-L|=|f(x,y)-10|=
3
x
y-
3
y
x
390
≤
3
x
y-
3
y
x≤
3
x
y+
3
y
x
by the triangle inequality, and

3
x
y+
3
y
x<
4
δ
+
4
δ
=2
4
δ
. If
1≤
ϵ
2
,
2
4
δ
=2≤ϵ
, and if
1>
ϵ
2
,
2
4
δ
<2δ=ϵ
. Thus by the choice of
δ
,
|f(x,y)-10|<ϵ
, and because
ϵ
is arbitrary, an appropriate
δ
can be found for any value of
ϵ
; hence the limit is 10.

External Links

Epsilon-Delta Definition (Wolfram MathWorld)
Limit (Wolfram MathWorld)

Permanent Citation

Spencer Liang
​
​"Multivariable Epsilon-Delta Limit Definitions"​
​http://demonstrations.wolfram.com/MultivariableEpsilonDeltaLimitDefinitions/​
​Wolfram Demonstrations Project​
​Published: January 11, 2008