L’Hôpital’s rule for indeterminate limits

E928482

calculus rule limit evaluation technique mathematical theorem

L’Hôpital’s rule for indeterminate limits is a fundamental calculus technique that evaluates certain indeterminate forms of limits by relating them to the limits of the derivatives of the functions involved.

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All labels observed (5)

Label	Occurrences
L’Hôpital’s rule	2
L’Hospital’s rule	1
L’Hospital’s rule for indeterminate limits	1
L’Hôpital’s rule for indeterminate limits canonical	1
l’Hôpital’s rule	1

How this entity was disambiguated

This entity first appeared as the object of triple T11478970 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.

NED1 Entity disambiguation (via context triple) gpt-5-mini-2025-08-07

Target entity: L’Hôpital’s rule for indeterminate limits
Context triple: [Guillaume de l’Hôpital, notableConcept, L’Hôpital’s rule for indeterminate limits]

A. Krak de l’Hospital
Krak de l’Hospital is an alternative name for Krak des Chevaliers, the famous medieval Crusader castle in Syria renowned for its massive fortifications and strategic importance.
B. Euler’s method of rearranging absolutely convergent series
Euler’s method of rearranging absolutely convergent series is a technique introduced by Leonhard Euler to systematically reorder and manipulate convergent infinite series in order to derive new identities and product expansions, such as those appearing in analytic number theory.
C. epsilon–delta definition of limit
The epsilon–delta definition of limit is the rigorous formalization of the intuitive notion of a function approaching a value, forming the foundation of modern analysis and calculus.
D. Cauchy–Hadamard theorem
The Cauchy–Hadamard theorem is a fundamental result in complex analysis that characterizes the radius of convergence of a power series in terms of the growth rate of its coefficients.
E. Essai sur l’étude des fonctions données par leur développement de Taylor
Essai sur l’étude des fonctions données par leur développement de Taylor is a foundational mathematical treatise by Jacques Hadamard that investigates the behavior and properties of functions defined through their Taylor series expansions.
F. None of above. chosen
G. Unsure - the case is ambiguous/there is not enough information to decide.

NED2 Entity disambiguation (via description) gpt-5-mini-2025-08-07

Target entity: L’Hôpital’s rule for indeterminate limits
Target entity description: L’Hôpital’s rule for indeterminate limits is a fundamental calculus technique that evaluates certain indeterminate forms of limits by relating them to the limits of the derivatives of the functions involved.

A. Krak de l’Hospital
Krak de l’Hospital is an alternative name for Krak des Chevaliers, the famous medieval Crusader castle in Syria renowned for its massive fortifications and strategic importance.
B. Euler’s method of rearranging absolutely convergent series
Euler’s method of rearranging absolutely convergent series is a technique introduced by Leonhard Euler to systematically reorder and manipulate convergent infinite series in order to derive new identities and product expansions, such as those appearing in analytic number theory.
C. epsilon–delta definition of limit
The epsilon–delta definition of limit is the rigorous formalization of the intuitive notion of a function approaching a value, forming the foundation of modern analysis and calculus.
D. Cauchy–Hadamard theorem
The Cauchy–Hadamard theorem is a fundamental result in complex analysis that characterizes the radius of convergence of a power series in terms of the growth rate of its coefficients.
E. Essai sur l’étude des fonctions données par leur développement de Taylor
Essai sur l’étude des fonctions données par leur développement de Taylor is a foundational mathematical treatise by Jacques Hadamard that investigates the behavior and properties of functions defined through their Taylor series expansions.
F. None of above. chosen

Statements (49)

Predicate	Object
instanceOf	calculus rule ⓘ limit evaluation technique ⓘ mathematical theorem ⓘ
alternativeName	L’Hospital’s rule ⓘ linked to: L’Hôpital’s rule for indeterminate limits
alternativeSpelling	L’Hospital’s rule for indeterminate limits ⓘ linked to: L’Hôpital’s rule for indeterminate limits
appliesTo	indeterminate form 0/0 ⓘ indeterminate form ∞/∞ ⓘ
assumes	real-valued functions in a neighborhood of the point ⓘ
canBeAppliedRepeatedly	true ⓘ
category	techniques for evaluating limits ⓘ
doesNotApplyTo	determinate limits ⓘ
field	calculus ⓘ mathematical analysis ⓘ
generalizationOf	Cauchy’s mean value theorem ⓘ
hasVariant	version for limits at infinity ⓘ version for one-sided limits ⓘ version for sequences via continuous extension ⓘ
historicalAttribution	first published in 1696 in l’Hôpital’s textbook "Analyse des Infiniment Petits" ⓘ
indirectlyUsedFor	indeterminate form 0^0 via logarithms ⓘ indeterminate form 0·∞ via algebraic manipulation ⓘ indeterminate form 1^∞ via logarithms ⓘ indeterminate form ∞^0 via logarithms ⓘ indeterminate form ∞−∞ via algebraic manipulation ⓘ
isGeneralizedBy	Cauchy’s mean value theorem proof ⓘ
isOftenMisusedBy	applying when hypotheses fail ⓘ
isTaughtIn	AP Calculus curriculum ⓘ introductory calculus courses ⓘ university analysis courses ⓘ
namedAfter	Guillaume de l’Hôpital ⓘ
proofUses	Cauchy mean value theorem ⓘ linked to: Cauchy’s mean value theorem mean value theorem ⓘ linked to: Lagrange’s mean value theorem
relatedTo	Taylor series methods for limits ⓘ asymptotic analysis ⓘ
relates	limit of a quotient to limit of derivative quotient ⓘ
requires	denominator derivative nonzero on a punctured neighborhood ⓘ existence of limit of derivative quotient or divergence to ±∞ ⓘ functions differentiable on an open interval around the point ⓘ
requiresCondition	derivative quotient limit must exist or be infinite ⓘ original limit must be in an indeterminate form ⓘ
statement	If lim_{x→a} f(x)=0 and lim_{x→a} g(x)=0 and f,g are differentiable near a with g′(x)≠0, then lim_{x→a} f(x)/g(x)=lim_{x→a} f′(x)/g′(x) when the latter limit exists or is infinite. ⓘ If lim_{x→a} \|f(x)\|=∞ and lim_{x→a} \|g(x)\|=∞ and f,g are differentiable near a with g′(x)≠0, then lim_{x→a} f(x)/g(x)=lim_{x→a} f′(x)/g′(x) when the latter limit exists or is infinite. ⓘ
typicalExample	lim_{x→0} (sin x)/x = 1 via derivatives cos x / 1 ⓘ lim_{x→∞} (ln x)/x = 0 via derivatives 1/x / 1 ⓘ
usedFor	evaluating difficult limits ⓘ resolving 0/0 indeterminate forms ⓘ resolving ∞/∞ indeterminate forms ⓘ
usesConcept	derivative ⓘ differentiability ⓘ limit ⓘ

How these facts were elicited