Limits

Topology Terminology

Let P be a point in the plane then a d-neighborhood (ball) of P is the set of points that are less than d units away from P.  If R is a region, then a point P is called an interior point of R if there is a d-neighborhood totally contained in R.  If every point of R is an interior point of R then R is called open. A point P is called a boundary point of R if every d-neighborhood of R contains both points in R and not in R.  R is called closed if it contains all of its boundary points.

The Definition of a Limit

 Definition Let f(x,y) be a function defined near the point P, then          if there is a d such that f(x,y) is close to L for all points (except possibly P) in the d - neighborhood of P.

Equivalently, the limit is L if for all paths that lead to P, the function also tends towards P.  (Recall that for the one variable case we needed to check only the path from the left and from the right.) To show that a limit does not exist at a point we need only find two paths that both lead to P such that f(x,y) tends towards different values.

Techniques For Finding Limits

Example

Show that

Does not exist

Solution

First select the path along the x-axis.  On this path

y = 0

so the function becomes:

0
f(x,0)  =              =  0
x2

Now choose the path along the y = x line:

x2            1
f(0,y)  =              =
2x2           2

Hence the function tends towards two different values for different paths.  We can conclude that the limit does not exist.  The graph is pictured below.

Example

Find

We could try the paths from the last example, but both paths give a value of 0 for the limit.  Hence we suspect that the limit exists.  We convert to polar coordinates and take the limit as r approaches 0:

We have

r3cos3q + r3sin3q
f(r,q)  =                                    =  rcos3q + rsin3q
r2

as r approaches 0 the function also approaches 0 no matter what q is.  Hence the limit is 0.

Below is the graph of this function.

Exercises:  Find the limit if it exists

Continuity

We make the following definition for continuity.

 Definition A function of several variables is continuous at a point P if the limit exists at P and the function defined at P is equal to this limit.

As with functions of one variable, polynomials are continuous, sums, products, and compositions of continuous functions are continuous.  Quotients of continuous functions are continuous.  A function is continuous if it is continuous at every point.

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