WOLFRAM|DEMONSTRATIONS PROJECT

Nonexistence of a Multivariable Limit

​
m
1
In multivariable calculus, a limit of a function
f(x,y)
exists at a point
(
x
0
,
y
0
)
if and only if we can make
f(x,y)
as close as we want to
f(
x
0
,
y
0
)
for all points
(x,y)
arbitrarily close to
(
x
0
,
y
0
).
One way to show that a limit does not exist (i.e. the definition fails) is to show that the function approaches different values from different directions. Akin to the notion of a one-sided limit in single-variable calculus, we consider the limit along a path in multivariable calculus. If the limit exists and is the same along all paths to
(
x
0
,
y
0
),
then the limit of the function exists.
This Demonstration gives an example where the limit does not exist. It examines the function
f(x,y)=
10xy
2
2
x
+3
2
y
, which is defined for all values of
(x,y)
in
2

except the origin. We consider the limit of the function as we approach the origin along the paths
y=mx
for all values of
m
between
-1
and
1
. Notice that the function approaches
-2
as
(x,y)
approaches the origin along the path
y=-x
, yet the function approaches
2
as
(x,y)
approaches the origin along the path
y=x.
As the limits along various paths do not agree, the limit of the function does not exist.