Generalizations of l’Hôpital’s rule

Abstract:

An essential limit, similar to the concept of essential bounded functions, is defined and briefly discussed. Using the essential limit, l’Hôpital’s rule is generalized to include the following theorem as a special case. Theorem. Let F, G be real-valued functions defined on the open interval (a, b). Suppose that the approximate derivatives ${F’_{{\text {ap}}}}(x)$ and ${G’_{{\text {ap}}}}(x)$ exist finitely, ${G’_{{\text {ap}}}}(x) > 0$ for almost all x in (a, b), and the extreme approximate derivates of both F and G are finite nearly everywhere in (a, b). Then ${\lim _{x \to a + }}[F(x)/G(x)] = {\text {ess}}\;{\lim _{x \to a + }}[{F’_{{\text {ap}}}}(x)/{G’_{{\text {ap}}}}(x)]$ provided that the essential limit in the right-hand side exists and that ${\lim _{x \to a + }}F(x) = {\lim _{x \to a + }}G(x) = 0$ or ${\lim _{x \to a + }}G(x) = - \infty$.