Exercise 9.  For  ,  let   .  Does    have a limit as  ?  

Solution 9.

See text and/or instructor's solution manual.

Answer.  No.  To see why, approach 0 along the real and imaginary axes respectively.

Solution.  

Along the real axis we have  ,  and

          

Along the imaginary axis we have  ,  and

          

Since the limits along the real and imaginary axes are different, we conclude that does not exist.

Aside.  Along the line    we have  ,   

and  .   So that different "limits" are obtained along different lines to the origin.  

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