L’Hôpital’s rule for indeterminate limits
E928482
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.
All labels observed (2)
| Label | Occurrences |
|---|---|
| L’Hôpital’s rule | 1 |
| L’Hôpital’s rule for indeterminate limits canonical | 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]
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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.
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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.
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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.
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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.
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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 NERFINISHED ⓘ |
| alternativeSpelling | L’Hospital’s rule for indeterminate limits NERFINISHED ⓘ |
| 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 NERFINISHED ⓘ |
| 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 NERFINISHED ⓘ |
| proofUses |
Cauchy mean value theorem
NERFINISHED
ⓘ
mean value theorem NERFINISHED ⓘ |
| 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
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Instruction
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Input
Subject: L’Hôpital’s rule for indeterminate limits Description of subject: 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.
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
L’Hôpital’s rule