In this insightful lecture, Ralph P. Boas, a well-known figure in American mathematics, brings more than 50 years of mathematical experience to bear on questions growing out of the seemingly simple theorem of l'Hôpital of elementary calculus. As he sketches some of the history of this famous result, he points to the advantages of various proofs and gives some wry commentary about why students prefer some proofs over others. Extending the discussion from Cauchy's proof of l'Hôpital's rule, Boas discusses the fact that the role of integration can be played by a general linear operator that is the inverse of a differential operator, such as a Laplace transform. He also points to a number of open topics of investigation in this area, showing how an elementary result like l'Hôpital's rule can lead to the frontiers of mathematical research. (Phoenix, AZ, 1989)