Exercise 15.  Let     Is    continuous at the origin?  

Solution 15.

See text and/or instructor's solution manual.

Answer.  No.  The limit does not exist.   Show why.

Solution.  We can compute the limits along the x-axis and y-axis and compare them.

The limit along the positive x-axis is:  

        .  


The limit along the negative x-axis is:  

        .  
        
The limit along the y-axis is:

        .  

Since these limits are different, we conclude that    is not continuous at  .   

We remark that the function can be written as:

        .  

We are done.   

Aside.  We can let Mathematica double check our work.

This solution is complements of the authors.

(c) 2008 John H. Mathews, Russell W. Howell