Example 2.15.  Show that the function    does not have a limit as (x,y) approaches .

Solution.  If we let (x,y) approach (0,0) along the x axis, then  

            .  

But if we let (x,y) approach (0,0) along the line , then  

            .  

Because the value of the limit differs depending on how (x,y) approaches  (0,0), we conclude that does not have a limit as approaches .

Explore Solution 2.15.

Enter the function  .  

Find the iterated limit,  .  

Find the iterated limit,  .  

Find the polar limit  .  

So, along all lines through the origin, the limit is 0.

The value of this limit is dependent on the angle  .  Therefore,     does not have a limit  as  (x,y)  approaches  (0,0).
We can use Mathematica to make a ContourPlot of  u(x,u).

Notice that the contours with the given "c" values all go through the origin.  Hence "many limits" are possible as one approaches the origin.  We see that    does not have a (unique) limit as  (x,y)  approaches  (0,0).

(c) 2006 John H. Mathews, Russell W. Howell