The proofs of these properties are similar to those for the limits of functions of one variable. We can apply these laws to finding limits of various functions.
Finding the limit of a function of two variables
Find each of the following limits:
First use the sum and difference laws to separate the terms:
Next, use the constant multiple law on the second, third, fourth, and fifth limits:
Now, use the power law on the first and third limits, and the product law on the second limit:
Last, use the identity laws on the first six limits and the constant law on the last limit:
Before applying the quotient law, we need to verify that the limit of the denominator is nonzero. Using the difference law, constant multiple law, and identity law,
Since the limit of the denominator is nonzero, the quotient law applies. We now calculate the limit of the numerator using the difference law, constant multiple law, and identity law:
Since we are taking the limit of a function of two variables, the point
is in
and it is possible to approach this point from an infinite number of directions. Sometimes when calculating a limit, the answer varies depending on the path taken toward
If this is the case, then the limit fails to exist. In other words, the limit must be unique, regardless of path taken.
Limits that fail to exist
Show that neither of the following limits exist:
The domain of the function
consists of all points in the
except for the point
(
[link] ). To show that the limit does not exist as
approaches
we note that it is impossible to satisfy the definition of a limit of a function of two variables because of the fact that the function takes different values along different lines passing through point
First, consider the line
in the
Substituting
into
gives
for any value of
Therefore the value of
remains constant for any point on the
and as
approaches zero, the function remains fixed at zero.
Next, consider the line
Substituting
into
gives
This is true for any point on the line
If we let
approach zero while staying on this line, the value of the function remains fixed at
regardless of how small
is.
Choose a value for
that is less than
—say,
Then, no matter how small a
disk we draw around
the values of
for points inside that
disk will include both
and
Therefore, the definition of limit at a point is never satisfied and the limit fails to exist.
Graph of the function
Along the line
the function is equal to zero; along the line
the function is equal to
In a similar fashion to a., we can approach the origin along any straight line passing through the origin. If we try the
(i.e.,
then the function remains fixed at zero. The same is true for the
Suppose we approach the origin along a straight line of slope
The equation of this line is
Then the limit becomes
regardless of the value of
It would seem that the limit is equal to zero. What if we chose a curve passing through the origin instead? For example, we can consider the parabola given by the equation
Substituting
in place of
in
gives
By the same logic in a., it is impossible to find a
disk around the origin that satisfies the definition of the limit for any value of
Therefore,
does not exist.
