Exercise 1.  Find the following limits.

1 (a).  .

Solution 1 (a).

See text and/or instructor's solution manual.

Answer   .

Solution.  Method I.  We have  

              

We can use the "squeeze theorem" for real sequences to show that both    and    converge to  0.

                    ,   and

                    ,  

Hence we have,       and    .

Therefore   .

Solution.  Method II.  Use the fact that      iff   .    (You will be asked to prove this fact in Exercise 17.)

Find the limit of the sequence of absolute values:

                      

Then,       implies that    .

We are done.   

Aside.  We can let Mathematica double check our work.

                    The graph of   ,   and the limit point   .  

This solution is complements of the authors.

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