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.
Back to the Functions of Several Variables Page
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