Example 1:

Here is a function of two variables that is not defined at zero:
(Note Abs[ ] is the function for absolute value

In[124]:=

In[125]:=

Out[125]=

Here mathematica computes the limit from the x-axis direction

In[126]:=

Out[126]=

and from the y-axis direction:

In[127]:=

Out[127]=

They are different--- This means the limit does not exist. Here is the contour plot:

In[128]:=

Out[128]=

Here is another contour plot, As we approach from the x-direction we see different colors then as we approach from the y direction

In[129]:=

Out[129]=

In[130]:=

Out[130]=

This plots the section of our surface lying above a strip along the x-axis

In[131]:=

Out[131]=

This plots the section of our surface lying above a strip along the y-axis

In[132]:=

Out[132]=

Example 2:

Here is a function of two variables that is not defined at zero:

In[133]:=

In[134]:=

Out[134]=

Here mathematica computes the limit from the x-axis direction

In[135]:=

Out[135]=

and from the y-axis direction:

In[136]:=

Out[136]=

But from the x=y line direction:

In[137]:=

Out[137]=

They are different--- This means the limit does not exist. Here is the contour plot:

In[138]:=

Out[138]=

In[139]:=

Out[139]=

In[140]:=

Out[140]=

In[141]:=

Out[141]=

Example 3

The following is an example where the function is not defined at (0,0) but the limit does exist

In[142]:=

In[143]:=

Out[143]=

Note: Mathematica won't take a square root of t^2!

In[144]:=

Out[144]=

However you can tell it to:

In[145]:=

Out[145]=

In[146]:=

Out[146]=

In[147]:=

Out[147]=

In[148]:=

Example 4

The following is a composition of a quotient of polynomials with a continuous function. Therefore it is continuous wherever it is defined. In this case, it is defined whenever x≠y.

In[149]:=

In[150]:=

Out[150]=

Here is a contour plot of this function:

In[151]:=

Out[151]=

Here is my attempt to pretty it up. Notice the contour plot "goes nuts" near the line x=y.

In[152]:=

Out[152]=

Here is the graph:
(The errors come from Mathematica trying to plot where the function is not defined)

In[153]:=

Out[153]=

Here we "zoom" in to the point (0,0)

In[154]:=

Out[154]=

Here is a pretty picture of the first graph:

In[160]:=

Out[160]=

Created by Mathematica  (September 23, 2004)