We investigate limits of functions of several variables.

*all*sufficiently close, but not equal to, .

This easily allows us to make a similar definition for functions of several variables.

**limit**of as approaches is if the value of can be made as close as one wishes to for all in the domain of sufficiently close, but not equal to, .

When this occurs, we write .

We now focus on functions of two variables, although it is not difficult to state similar results in the more general setting. For the rest of what follows, we will often denote points by .

While the intuitive idea behind limits seems to remain unchanged, something interesting is worth observing. One of the most important ideas for limits of a function of a single variable is the notion of a sided limit. For functions of a single variable, there were really only two natural ways for to become close to ; we could take to approach the point from the left or the right. For instance, tells us to consider the inputs only. In fact, there’s a theorem that guarantees that if and only if and , meaning that the function must approach the same value as the input approaches from both the left and the right.

On the other hand, there are now *infinitely many* ways for ; we can approach along a
straight line path parallel to the -axis or -axis, other straight line paths, or even other
types of curves.

In order to check whether a limit exists, do we have to verify that the function tends to the same value along infinitely many different paths?

While this may seem problematic, there is some good news; many of the limit laws from before still do hold now.

- Constant Law
- Identity Law
- , where
- Sum/Difference Law
- Scalar Multiple Law
- Product Law
- Quotient Law
- , if

In practice, this allows us to compute many limits in a similar fashion as before.

Essentially, the above laws allow us to evaluate limits by directly substituting values into the given function, provided the end result is a constant. Henceforth, when a limit can be evaluated by direct substitution, we will not show the details.

As it turns out, another old technique works well too.

What allows us to perform the cancellation of the common factors of ? Note that when determining whether a limit exists or not, we must look near the point , but not at the point . No matter how close a point is to the point , as long as , then . So, this cancellation is valid.

### When limits don’t exist

Unfortunately, there are difficulties that arise now that did not before when we have
to handle indeterminate forms. Since limits exist only when the function tends to the
same value along *every* path, we can use this to show that some limits do not
exist.

**If it is possible to arrive at different limiting values by
approaching along different paths, the limit does not exist.**

This is analogous to the left and right hand limits of single variable functions not being equal, implying that the limit does not exist.

For , note that *any* path that approaches must eventually lie in the portion of the
domain of where . The image below shows the domain of and the formulas used to
evaluate it for each in the -plane.

We see that for any point sufficiently close to the following holds

Now, let’s consider an example in which we do not have a piecewise function.

- If we approach along the line in the -plane, we have Note that this tells
us that
*if*the limit exists, it*must*be . - If we approach along the line in the -plane, we have Note that this tells
us that
*if*the limit exists, it*must*be .

Since approaches different values along different paths,

does not exist.

As it turns out, there are nice ways to think about the above result both geometrically and analytically.

**A geometric viewpoint of analyzing along level curves**

These level curves share as a boundary point; had either been defined there, they would intersect. Since there are different -values associated to each level curve, it is impossible that exists. This notion can be generalized.

**If two different level curves of a function share a common
boundary point, then the limit at that boundary point does
not exist.**

**An analytic viewpoint of analyzing along level curves**

Note that this “cancellation” is the algebraic consequence of analyzing along a level
curve; in order for the function to be constant, there can be no explicit dependence
on and . However, the function tends to a *different* value as along for each choice
of , so the limit cannot exist.

*class*of paths along which the function tends to different values. This occurs because the lines are all parts of different level curves of .

We finish by presenting you with a plot of :

We conclude this section with an example that illustrates the necessity to analyze a
function along *every* path in its domain in order to conclude that a limit
exists.

*except*when , that is, along the line . This line is not in the domain of , so we have found the following fact: along every line in the domain of , Now consider the limit along the path : Now apply L’Hôpital’s Rule twice to find a limit is of the form . Hence the limit does not exist. Step back and consider what we have just discovered.

- Along any line in the domain of the , the limit is .
- However, along the path , which lies in the domain of the for all , the limit does not exist.

Since the limit is not the same along every path to , we say does not exist. We finish by presenting you with a plot of :