Solution 3 (g).

See text and/or instructor's solution manual.

Answer.      has a removable singularity at the origin if we define  .

Solution.   Consider  .  Use the known series    and write

                    .

Substitute this series in the numerator and obtain

                       

Then apply  Definition 7.5 to conclude that     has a removable singularity at the origin.

We are  done.   

Aside.  We can let Mathematica double check our work.

We are really done.   

                              The removable singularity at the origin.

This solution is complements of the authors.

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