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Limits and Continuity of Two Dimensional Functions

Objectives

In this lab you will use the Mathematica to get a visual idea about the existence and behavior of limits of functions of two variables. You will also begin to use some of Mathematica's symbolic capacities to advantage.

The Two Functions

We will compare and contrast two functions with respect to their behavior at x=0=y,

displaymath220

First, input these two functions into Mathematica by defining functions as follows:

       f[x_,y_] := x y /(x^2 + y^2)
The square brackets, the underscores, and the `:=' are necessary. The advantage of defining a function in Mathematica is that you can now evaluate it for a variety of arguments. For example, to see what the function looks like along the curve tex2html_wrap_inline224 , type
       f[x, x^2+7]
and shift-return. To evaluate the function at the point (1,3), simply type
       f[1,3]
and shift-return.

Graphics for Functions

Produce contour and surface plots of the functions f and g. Use the following format for commands:

       Plot3D[f[x,y], {x,-2,2}, {y,-2,2}];
       Plot3D[g[x,y], {x,-2,2}, {y,-2,2}];
with whatever options you want. Since f and g are now defined as functions, the variables must be included explicitly in the plotting commands. Compare the contours of the two functions and the surface plots of the two functions - write a note below the pairs of graph describing how the pictures you are generating relate to the existence and nonexistence of the limit at zero.

Distinguished Limits

A powerful technique for examining the behavior of limits in many dimensions is to look at one-dimensional limits along parametric curves. Sometimes these are called `distinguished limits.' Mathematica is particularly useful for doing this. For example, suppose we want to examine the behavior of g(x,y) along the line y=3x. We can think of this line as being parametrized by x=t and y=3t. Then, along the line g(t) = g(t,3t) is a function of only one variable, t, and we know a lot about functions of one variable. For a function of many variables to have a limit at a point, the limit of every parametric curve passing through the point must agree.

In Mathematica it is easy to examine distinguished limit. The sequence of commands below plots a graph of g(t) as described above.

       Plot[g[t, 3t], {t,-1,1}, AxesLabel->{"t","g(t)"}];
Here is how to plot two (or more) such graphs simultaneously:
       Plot[{g[t, 3t], g[t, .5 t]}, {t,-1,1}, AxesLabel->{"t","g(t)"}];
Notice that the distinguished limits of g (which we know to have a limit at zero) agree at t=0. Perform some similar calculations to illustrate the nonexistence of the limit of f at zero.

Finally

Pick a function from your homework whose limit/continuity behavior seems perplexing to you and use some of the commands in the preceding section to examine its behavior.


next up previous contents
Next: Partial DerivativesDifferentials, and Up: Computational Labs in Mathematica Previous: Graphic Representations of Two

James Powell
Fri Feb 14 12:47:33 MST 1997