Solution 5.

Solution.   For the real function    and  ,  we have

          .  

This implies that   .  

Therefore, the real function   is continuous at  .  

      The complex function      has an essential singularity at the origin.  

Use the fact that  ,  and express this function as a Laurent series.  

Use the substitution    and get  

                      

Then apply Definition 7.5 to conclude that      has an essential singularity at the origin.

Therefore, the complex function    is not continuous at  .  

We are done.   

Aside.  We can let Mathematica double check our work.

The limit for the real function   is  .  

There is not a unique limit for the complex function    as  .  

For example, the limits along the real and imaginary axis are different:

                    ,
but
                    .  

We can set    then   .  

Now the limit along the imaginary axis is easier to compute

                      

Therefore, the complex function    is not continuous at  .  

          The graphs of      and    .

                    A plot for   .                                 A plot for   .  

This solution is complements of the authors.

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