In this section we will take a look at limits involving
functions of more than one variable. In
fact, we will concentrate mostly on limits of functions of two variables, but
the ideas can be extended out to functions with more than two variables.
Before getting into this let’s briefly recall how limits of
functions of one variable work. We say
that,
provided,
Also, recall that,
is a right hand limit and requires us to only look at values
of x that are greater than a.
Likewise,
is a left hand limit and requires us to only look at values
of x that are less than a.
In other words, we will have provided approaches L
as we move in towards (without letting ) from both sides.
Now, notice that in this case there are only two paths that
we can take as we move in towards . We can either move in from the left or we can
move in from the right. Then in order
for the limit of a function of one variable to exist the function must be
approaching the same value as we take each of these paths in towards .
With functions of two variables we will have to do something
similar, except this time there is (potentially) going to be a lot more work
involved. Let’s first address the
notation and get a feel for just what we’re going to be asking for in these
kinds of limits.
We will be asking to take the limit of the function as x
approaches a and as y approaches b. This can be written in
several ways. Here are a couple of the
more standard notations.
We will use the second notation more often than not in this
course. The second notation is also a
little more helpful in illustrating what we are really doing here when we are
taking a limit. In taking a limit of a
function of two variables we are really asking what the value of is doing as we move the point in closer and closer to the point without actually letting it be .
Just like with limits of functions of one variable, in order
for this limit to exist, the function must be approaching the same value
regardless of the path that we take as we move in towards . The problem that we are immediately faced
with is that there are literally an infinite number of paths that we can take
as we move in towards . Here are a few examples of paths that we
could take.
We put in a couple of straight line paths as well as a
couple of “stranger” paths that aren’t straight line paths. Also, we only included 6 paths here and as
you can see simply by varying the slope of the straight line paths there are an
infinite number of these and then we would need to consider paths that aren’t
straight line paths.
In other words, to show that a limit exists we would
technically need to check an infinite number of paths and verify that the
function is approaching the same value regardless of the path we are using to
approach the point.
Luckily for us however we can use one of the main ideas from
Calculus I limits to help us take limits here.
Definition
From a graphical standpoint this definition means the same
thing as it did when we first saw continuity
in Calculus I. A function will be
continuous at a point if the graph doesn’t have any holes or breaks at that point.
How can this help us take limits? Well, just as in Calculus I, if you know that
a function is continuous at then you also know that
must be true. So, if
we know that a function is continuous at a point then all we need to do to take
the limit of the function at that point is to plug the point into the function.
All the standard functions that we know to be continuous are
still continuous even if we are plugging in more than one variable now. We just need to watch out for division by
zero, square roots of negative numbers, logarithms of zero or negative numbers,
etc.
Note that the idea about paths is one that we
shouldn’t forget since it is a nice way to determine if a limit doesn’t exist. If we can find two paths upon which the
function approaches different values as we get near the point then we will know
that the limit doesn’t exist.
Let’s take a look at a couple of examples.
Example 1 Determine
if the following limits exist or not.
If they do exist give the value of the limit.
(a) [Solution]
(b) [Solution]
(c) [Solution]
(d) [Solution]
Solution
(a)
Okay, in this case the function is continuous at the point
in question and so all we need to do is plug in the values and we’re done.
[Return to Problems]
(b)
In this case the function will not be continuous along the
line since we will get division by zero when this
is true. However, for this problem
that is not something that we will need to worry about since the point that
we are taking the limit at isn’t on this line.
Therefore, all that we need to do is plug in the point
since the function is continuous at this point.
[Return to Problems]
(c)
Now, in this case the function is not continuous at the
point in question and so we can’t just plug in the point. So, since the function is not continuous at
the point there is at least a chance that the limit doesn’t exist. If we could find two different paths to approach
the point that gave different values for the limit then we would know that
the limit didn’t exist. Two of the
more common paths to check are the x
and yaxis so let’s try those.
Before actually doing this we need to address just what
exactly do we mean when we say that we are going to approach a point along a
path. When we approach a point along a
path we will do this by either fixing x
or y or by relating x and y through some function.
In this way we can reduce the limit to just a limit involving a single
variable which we know how to do from Calculus I.
So, let’s see what happens along the xaxis. If we are going to approach along the xaxis we can take advantage of the fact that that along the xaxis we know that . This means that, along the xaxis, we will plug in into the function and then take the limit as
x approaches zero.
So, along the xaxis
the function will approach zero as we move in towards the origin.
Now, let’s try the yaxis. Along this axis we have and so the limit becomes,
So, the same limit along two paths. Don’t misread this. This does NOT say that the limit exists and
has a value of zero. This only means
that the limit happens to have the same value along two paths.
Let’s take a look at a third fairly common path to take a
look at. In this case we’ll move in
towards the origin along the path . This is what we meant previously about
relating x and y through a function.
To do this we will replace all the y’s with x’s and then
let x approach zero. Let’s take a look at this limit.
So, a different value from the previous two paths and this
means that the limit can’t possibly exist.
Note that we can use this idea of moving in towards the
origin along a line with the more general path if we need to.
[Return to Problems]
(d)
Okay, with this last one we again have continuity problems
at the origin. So, again let’s see if
we can find a couple of paths that give different values of the limit.
First, we will use the path . Along this path we have,
Now, let’s try the path . Along this path the limit becomes,
We now have two paths that give different values for the
limit and so the limit doesn’t exist.
As this limit has shown us we can, and often need, to use
paths other than lines.
[Return to Problems]
