L'Hôpital's rule

A theorem that evaluates certain indeterminate limits of quotients by taking the limit of the ratio of their derivatives, when its hypotheses are satisfied.

Indeterminate form: An expression whose form alone does not determine the value of a limit. The rule applies to quotient limits with forms such as 0/0 or ∞/∞.
Limit of a function: The value a function's outputs approach as its input approaches a specified point or infinity. The theorem is a method for finding particular limits of quotients.
Limit at infinity: The behavior of a function as its input grows without bound in a specified direction. L'Hôpital's rule often compares growth rates in quotients as the input tends to infinity.
Algebraic simplification: Rewriting an expression into an equivalent form that is easier to evaluate or analyze. Factoring or cancellation may solve a limit more directly than differentiating numerator and denominator.
Guillaume de l'Hôpital: A French mathematician who published the first calculus textbook and the rule now bearing his name. His 1696 textbook presented the result, though its derivation came from Johann Bernoulli.
Cauchy's mean value theorem: A theorem equating a ratio of function increments to a ratio of derivatives at an intermediate point. It provides the central justification for transferring a quotient limit to derivatives.
Derivative: The instantaneous rate of change of a function with respect to its input. The rule replaces a quotient of functions with a quotient of their derivatives.
Asymptotic analysis: The study of how functions behave relative to one another near a limit or at large inputs. Derivative ratios can establish which terms dominate in a quotient's limiting behavior.
Direct substitution: Evaluating a function at the limiting input when continuity makes that evaluation valid. When substitution gives a determinate quotient, the theorem is unnecessary.
Johann Bernoulli: A Swiss mathematician whose work helped establish calculus and included the rule later named for l'Hôpital. Bernoulli communicated the rule to l'Hôpital and supplied its original derivation.

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Stigler's law of eponymyRelated: Guillaume de l'Hôpital published the rule after learning it from Johann Bernoulli.