Theorem 3.2, (L'Hôpital's Rule).  Assume that f(z) and g(z) are both analytic at .  If , , and   then  

            .  

Exploration for L'Hospital's Rule.

Mathematica can do it all, however care must be taken with the higher derivative cases.  Make the assumption that  .  

Now, make the second additional assumption that  .  

The third additional assumption is that  .  

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