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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Observed surface forms (1)

Surface form	Occurrences
L’Hôpital’s rule	1

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 ⓘ

Referenced by (2)

Full triples — surface form annotated when it differs from this entity's canonical label.

Guillaume de l’Hôpital → notableConcept → L’Hôpital’s rule for indeterminate limits ⓘ

Guillaume de l’Hôpital → ruleNamedAfter → L’Hôpital’s rule for indeterminate limits ⓘ

this entity surface form: L’Hôpital’s rule