L'Hôpital's rule

alternative L'Hôpital's Rule · alternative l'Hôpital's Rule · alternative l'Hôpital's rule · alternative L'Hospital's rule · alternative rule of L'Hôpital · alternative rule of l'Hôpital

1. mathematicsThe rule that the limit of the ratio of two functions equals the limit of the ratio of their derivatives, usable when the former limit is indeterminate and the latter limit exists.

1. Alternative letter-case form of L'Hôpital's rule.

   In each of the examples presented in this section, L’Hôpital’s Rule is used to find a limit that exists.

   2011, Ron Larson, Bruce H. Edwards, David C. Falvo, “Appendix B: Additional Topics”, in Calculus I with Precalculus: A One-Year Course, 3rd edition, Brooks/Cole, 1 (L’Hôpital’s Rule), page A23

   2 more examples

   In this section, we shall describe a useful method for finding limits that is known as L’Hôpital’s Rule. Actually, there are several versions of L’Hôpital’s Rule and the formal statements of these will appear in the form of propositions or as a part of some remarks.

   2006, Sudhir R. Ghorpade, Balmohan V. Limaye, “Differentiation”, in A Course in Calculus and Real Analysis, Springer, →ISBN, section 4 (L’Hôpital’s Rule), page 131

   Modern examples involving L’Hôpital’s Rule frequently involve transcendental functions, but Leibniz’ calculus at this time was only a calculus of algebraic expressions.

   2015, Robert E. Bradley, Salvatore J. Petrilli, C. Edward Sandifer, “Translators’ Preface”, in L’Hôpital’s Analyse des infiniments petits: An Annotated Translation with Source Material by Johann Bernoulli, Birkhäuser, →ISBN, page xiv

1. Alternative letter-case form of L'Hôpital's rule.

   To resolve a limit involving an indeterminate form, we sometimes need to apply l’Hôpital’s Rule more than once.

   2017, Soo T. Tan, “Appendix B”, in Applied Calculus for the Managerial, Life, and Social Sciences, 10th edition, Cengage Learning, section 2 (The Indeterminate Forms 0/0 and ∞/∞ and l’Hôpital’s Rule), page 845

   2 more examples

   Many times a simple algebraic manipulation will put a limit into a form that we can study using l’Hôpital’s Rule.

   2006, Brian E. Blank, Steven George Krantz, “Applications of the Derivative”, in Calculus: Single Variable, Key College Publishing, →ISBN, section 6 (L’Hôpital’s Rule), page 303

   Notice that if you put x = 3, then both top and bottom of the fraction are 0. This means we can use l’Hôpital’s Rule.

   2007, Adrian Banner, “L’Hôpital’s Rule and Overview of Limits”, in The Calculus Lifesaver: All the Tools You Need to Excel at Calculus, Princeton, N.J.; Oxford: Princeton University Press, →ISBN, section 1 (L’Hôpital’s Rule), subsection 1 (Type A: 0/0 case), page 295

1. Alternative letter-case form of L'Hôpital's rule.

   We will discuss the proof of l’Hôpital’s rule later in this section.

   2016, Daniel J. Velleman, “Applications of Differentiation”, in Calculus: A Rigorous First Course, Mineola, N.Y.: Dover Publications, Inc., →ISBN, section 8 (L’Hôpital’s Rule), page 262

   2 more examples

   Many times, the limit of an exponential form can be determined by applying l'Hôpital's rule to the logarithm of the desired function.

   1985, Frederick H. Soon, “Limits, L'Hôpital's Rule, and Numerical Methods”, in Student’s Guide to Calculus by J. Marsden and A. Weinstein, volume II, Springer-Verlag, →ISBN, section 2 (L'Hôpital's Rule), page 484

   There are a number of applications of limits in physical chemistry, and l’Hôpital’s rule is useful in some of them.

   2005, Robert G. Mortimer, “Mathematical Functions and Differential Calculus”, in Mathematics for Physical Chemistry, 3rd edition, Elsevier, section “Limiting Values of Functions: L’Hôpital’s Rule”, page 114