Exercise 1.  Find the following limits.  

1 (c).  .  

Solution 1 (c).

See text and/or instructor's solution manual.

Answer.  .

Solution.  Substituting    will result in the "" situation.  

        

        Indeterminate

We can factor the numerator and denominator and get:  

        

We are done.   

Aside.  We can let Mathematica double check our work.

                    The mapping  ,  as  ,  the image point  ,

                    and we see that   limit   .  

Alternate Solution.  In Section 7.5 we will present the conditions under which L'Hôpital's rule is valid.  For most practical purposes it will appear similar to the rule learned in calculus.

           

Remark.  For this type of problem L'Hôpital's rule is faster.

This solution is complements of the authors.

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