Contents      [PD.1] [PD.2][PD.3] [PD.4] [PD.5] [PD.6] [PD.7] [PD.8]

PD.2 Limits and Continuity

We used limits in 2-D to help us determine what value in the range a number was approaching. We do the same thing in 3-D, we use the equation to help us find out what number the graph is approaching. However, it is a little more complicated because we are dealing with three dimensions rather than two. Here is the definition for a limit in 3-D:

 

Definition:

 

Let $f$ be a function of two variables whose domain $D$ includes points arbitrarily close to $(a,b)$. Then we say that the limit of $f(x,y)$

as $(x,y)$ approaches $(a,b)$ is $L$ and we write:


MATH

 

if for every number $\varepsilon >0$ there is a corresponding number $\delta >0$ such that


MATH

Above the limit is figured by taking the limit of the function along only one path. This is sufficient in 2-D, however it is not sufficient in 3-D. This is because a person can take more than one path of approach to a point. This idea is given below in a formal definition:

 

Definition:

If MATH as MATH along a path C $_{1}$ and MATH as MATH along a path C $_{2},$ where $L_{1}\neq L_{2}$, then
MATH does not exist.

 

 

 

 

picture of limit that would work with one path, but doesn't work with 2 paths

 

 

 

 

Now lets look at limits that do exist. Just as for functions of one variable, the calculation of limits for functions of two variables can be greatly simplified by the use of properties of limits. The Limit laws listed in Section 2.3 can be extended to functions of two variables. The limit of a sum is the sum of the limits, the limit of a product is the product of the limits, and so on. In particular, the following equations are true:

 
MATH
 

The Squeeze Theorem also holds.

 

Continuity

Recall that evaluating limits of continuous functions of a single variable is easy. It can be accomplished by direct substitution because the defining property of a continuous function is MATH. Continuous functions of two variables are also defined by the direct substitution property.

 

Definition

A function $f$ of two variables is called continuous at $(a,b)$ if

MATH
 

We say $f$ is continuous on $D$ if $f$ is continuous at every point $(a,b)$ in $D$.


Functions with Three or More Variables

 

Everything that we have done in this section so far can be extended into three or more variables. For example, a limit with three variables looks like this:


MATH

 

This means that the values of $f(x,y,z)$ approach the number $L$ as the point ($x,y,z)$ approaches the point ($a,b,c)$ along any path in the domain of $f$. The function $f$ is continuous at $(a,b,c)$ if


MATH







maybe a picture of something in 3 variables (its up to you)





In more than two variables, the formal definition of a limit looks like this:


If $f$ is defined on a subset $D$ of $\U{211d} ^{n}$, then MATH means that for every number $\epsilon >0$ there is corresponding number $\delta >0$ such that  MATH whenever x $\in D $ and MATH
 



 Contents      [PD.1] [PD.2][PD.3] [PD.4] [PD.5] [PD.6] [PD.7] [PD.8]

This document created by Scientific WorkPlace 4.1.