Exercise 7 (b).  Find   .  

Solution 7 (b).

Answer.   .  

Solution.  Recall that      so that

                    .

Since       and       we have

                       as   ,    and       as   .

Then  
                     
                        as   .

                        as   .

                        as   .

                        as   .

                        as   .

                        as   .

We are done.   

If we are careful to observe that     then we can consider the following limit argument.

                      

We are really done.   

Aside.  We can look at a graph.

                    The curve      is asymptotic to the line   .  

                    In quadrant III we have   .

We are really really done.   

Aside.  We can let Mathematica double check our work.

In this case we use the two variable formula     to get the correct answer.

You will get a wrong answer if you use the one variable formula   .

The answer    is wrong because the points    lie in quadrant III when   .

Indeed the curve     is asymptotic to the line  .

This solution is complements of the authors.

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