Example 2.19.  Show that .  

Solution.  Here P and Q can be factored in the form  

            

so that the limit is obtained by the calculation  

               

Explore Solution 2.19.

Enter the function f[z] and find the limit.

Try to evaluate f[1+i].

Let Mathematica take the limit.

Check out the details.
Factor the numerator and denominator of  f[z].

Find the limit of f[z] as z approaches 1 + i.

We can use Mathematica to make a mapping of a neighborhood of    in the z-plane and its image in the w-plane.

We see that  .  

(c) 2006 John H. Mathews, Russell W. Howell