I'll give the precise definition of a limit so that you can see the similarity to the definition you saw in single-variable calculus. The first definition is a technical point which you don't need to worry about too much. It simply ensures if we take a limit as , that x can approach c through a set where the function is defined.

* Definition.* Let U be a subset of . A point is an * accumulation point* of U if for every , the open ball contains a point
other than c.

is the set of points in which are less than r units from c:

* Definition.* Let be a function defined on , and let c be an accumulation point of U.
Then means:

For every , there is a , such that

Many results you know about limits from single-variable calculus have analogs for functions of several variables.

* Proposition.* Suppose where . Let c be an accumulation point of U, and
let . Then:

(a) .

(b) .

(c) .

(d) , provided that .

All of these results mean that if the limits on the right side are defined, then so is the limit on the left side, and the two sides are equal.

I won't try to state all of the easy results on limits that generalize to functions of several variables. You will see many of them proven in a course in real analysis. Let's look at some of the complications that result from being able to approach a point in more than one dimension.

* Example.* Compute .

I can compute the limit by plugging in. (This is another way of saying that is continuous at .) Thus.

* Example.* Compute
.

I can compute the limit by plugging in.

* Example.* Compute .

Substituting yields the indeterminate form .

Here's the graph of the function. Notice that as the height seems to approach one value along the "ridgeline" while it approaches another value along the "valley":

This leads me to believe that the limit is undefined.

To prove this, I try to find different ways of approaching which give different limits. Specifically, I try to pick different curves through which make simplify to different values. In this case, I try the x-axis, which is , and the y-axis, which is .

Set and let . I have

Set and let . I have

Since approaches different numbers depending on how approaches , the limit is undefined.

* Example.* Compute .

Substituting yields the indeterminate form .

Here's the graph of the function. It is a little harder to tell from the graph what is happening near the origin.

It turns out that the limit is undefined. To show this, I'll approach along a line and along a curve.

If you approach along the line , you get

Next, I notice that and . Thus, I can get multiple " " terms by setting .

If you approach along the curve , you get

Since approaches different numbers depending on how approaches , the limit is undefined.

* Example.* Compute by converting to polar coordinates.

Let and . Then

Then

* Example.* Compute .

I try to find different curves through which make simplify to different values.

If you approach along the line , , you get

If you approach along the line , you get

Since you get different limits by approaching in different ways, is undefined.

* Definition.* Let , where , and let . Then f is * continuous* at c if

* Remark.* Some authors will say a function is not continous at a point where
the function isn't defined. For example, the function f defined by
has (natural) domain . These authors will say that f is not continuous at
, with similar terminology for
multivariable functions. I'll avoid doing this: It seems
inappropriate to talk about whether a function does or does not have
a property like continuity at the point where there is no function!

I'll only consider continuity (or lack of continuity) at points in a function's domain.

* Example.* A function is defined by

Is f continuous at ?

However, ,

Since , it follows that f is not continuous at .

Copyright 2018 by Bruce Ikenaga