Exercise 7.  Does      exist?   Why?  

Solution 7.

See text and/or instructor's solution manual.

Answer   No, the limit does not exist.   

Solution.   We can use polar coordinates and write   .  

Then use De Moivre's Formula that was introduced in Section 1.5 and obtain:

                    .  

Since      and      do not exist, we conclude that      does not exist.  

We are done.   

Indeed, the sequence   cycles around the eighth roots of unity (see Example 1.19 in Section 1.5),  

                    .  

Therefore,      does not exist.  

We are really done.   

Aside.  We can let Mathematica double check our work.

                    The graph of   .   

                    This sequence continues to cycle around these eight points.  There is no limit for this sequence!

This solution is complements of the authors.

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