Name: limits of zero times infinity
Uploaded: 2026-01-12T12:24:55.000Z
Duration: 1 min 29 s
Channel: JoVE

Des formes indéterminées peuvent apparaître dans des limites de produits, où un facteur tend vers zéro, et l’autre tend vers plus ou moins infini.Le résultat est indéterminé ou incertain. Cela peut être nul, infini ou un nombre fini.Par exemple, considérons la limite unilatérale de la fonction produit suivante.Lorsque x tend vers zéro, le premier facteur x diminue à zéro, tandis que le second facteur, le logarithmide naturel de un sur x, tend vers l’infini.Le produit est algébriquement égal à un quotient qui peut s’écrire en deux expressions différentes.La première expression crée une forme infini sur l’infini, tandis que la seconde crée une forme zéro sur zéro.La règle de L’Hôpital peut s’appliquer à l’une ou l’autre forme. Mais, utiliser la première forme simplifie la différentiation et donne zéro en conséquence.Une visualisation utile est un polygone régulier inscrit dans un cercle. À mesure que le nombre de côtés augmente à l’infini, chaque côté diminue vers zéro.Cela crée un produit indéterminé : des faces infinies multipliées par zéro longueur. Pourtant, le périmètre total approche la circonférence fixe du cercle.Cela montre comment une forme nulle fois l’infini peut produire un résultat fini et significatif.

produits indéterminés

Indeterminate Products

Des formes ind&#xE9;termin&#xE9;es apparaissent &#xE9;galement lors de l&#x2019;&#xE9;valuation de limites portant sur des produits, en particulier lorsqu&#x2019;un facteur tend vers z&#xE9;ro tandis que l&#x2019;autre tend vers l&#x2019;infini positif ou n&#xE9;gatif. Cette situation, commun&#xE9;ment qualifi&#xE9;e de forme z&#xE9;ro fois l&#x2019;infini, ne poss&#xE8;de pas de r&#xE9;sultat imm&#xE9;diatement interpr&#xE9;table. Selon le comportement relatif des facteurs l&#x2019;un par rapport &#xE0; l&#x2019;autre, la limite d&#x2019;un tel produit peut &#xEA;tre nulle, infinie ou &#xEA;tre une valeur finie non nulle.
Limites de produits et r&#xE9;&#xE9;criture alg&#xE9;brique
Pour analyser des limites de ce type, il convient tout d&#x2019;abord de r&#xE9;&#xE9;crire le produit sous la forme d&#x2019;un quotient afin d&#x2019;appliquer la r&#xE8;gle de L&#x2019;H&#xF4;pital. Consid&#xE9;rez une limite unilat&#xE9;rale dans laquelle la variable tend vers z&#xE9;ro : l&#x2019;un des facteurs d&#xE9;cro&#xEE;t alors vers z&#xE9;ro, tandis que l&#x2019;autre facteur, par exemple le logarithme n&#xE9;p&#xE9;rien de l&#x2019;inverse de la variable, cro&#xEE;t sans borne. Bien que l&#x2019;expression initiale soit un produit, elle peut &#xEA;tre r&#xE9;&#xE9;crite alg&#xE9;briquement sous deux formes de quotient &#xE9;quivalentes. L&#x2019;une des r&#xE9;&#xE9;critures conduit &#xE0; la forme PT_19834_1, tandis que l&#x2019;autre conduit &#xE0; la forme PT_19834_2. Ces deux repr&#xE9;sentations sont valides et autorisent l&#x2019;application de la r&#xE8;gle de L&#x2019;H&#xF4;pital.
L&#x2019;application de cette r&#xE8;gle consiste &#xE0; d&#xE9;river s&#xE9;par&#xE9;ment le num&#xE9;rateur et le d&#xE9;nominateur, puis &#xE0; &#xE9;valuer la limite obtenue. Bien que l&#x2019;une ou l&#x2019;autre des formes du quotient puisse &#xEA;tre utilis&#xE9;e, le choix de la forme peut influer sur la simplicit&#xE9; de la d&#xE9;rivation. Dans le cas pr&#xE9;sent, la forme de type infini sur infini conduit &#xE0; une d&#xE9;rivation plus simple, mettant en &#xE9;vidence que la limite du produit initial est nulle.
Interpr&#xE9;tation g&#xE9;om&#xE9;trique
Une analogie g&#xE9;om&#xE9;trique utile pour comprendre les formes ind&#xE9;termin&#xE9;es de type z&#xE9;ro fois l&#x2019;infini est celle d&#x2019;un polygone r&#xE9;gulier inscrit dans un cercle. Lorsque le nombre de c&#xF4;t&#xE9;s du polygone augmente sans borne, la longueur de chaque c&#xF4;t&#xE9; individuel tend vers z&#xE9;ro. Le p&#xE9;rim&#xE8;tre total, en revanche, est le produit du nombre de c&#xF4;t&#xE9;s par la longueur de chacun d&#x2019;eux. On obtient ainsi un produit ind&#xE9;termin&#xE9; faisant intervenir une infinit&#xE9; de c&#xF4;t&#xE9;s dont la longueur tend vers z&#xE9;ro. Malgr&#xE9; cela, le p&#xE9;rim&#xE8;tre tend vers une valeur finie et bien d&#xE9;finie, &#xE9;gale &#xE0; la circonf&#xE9;rence du cercle. Cet exemple illustre la mani&#xE8;re dont des formes z&#xE9;ro fois l&#x2019;infini peuvent conduire &#xE0; des limites finies et non nulles lorsqu&#x2019;elles sont analys&#xE9;es de fa&#xE7;on appropri&#xE9;e.

Produits indéterminés

Des formes ind&#xE9;termin&#xE9;es apparaissent &#xE9;galement lors de l&#x2019;&#xE9;valuation de limites portant sur des produits, en particulier ...

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Calculus

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Applications of Differentiation

Applications de la dérivation

KEY TERMS AND CONCEPTS

TERMES ET CONCEPTS CLÉS

limits of zero times infinity

Limits with a zero-times-infinity product are indeterminate. In this type of limit, one factor approaches zero while the other factor grows toward positive or negative infinity. The result is not obvious at first, because the product may approach zero, infinity, or a finite nonzero value.

To evaluate these product limits, the expression is first rewritten as a quotient. This form makes it possible to use L'Hôpital's Rule, which compares the derivatives of the numerator and denominator. A one-sided limit can be rewritten in more than one equivalent quotient form. Either form is valid, but one may be easier to differentiate than the other.

In the example shown, the form with infinity over infinity gives a simpler derivative. After applying L'Hôpital's Rule, the limit of the original product is zero. This shows that a zero-times-infinity expression can have a definite limit once it is rewritten and analyzed correctly.

A geometric example helps make the idea clearer. A regular polygon inscribed in a circle has sides that get shorter as the number of sides increases without bound. At the same time, the number of sides grows without bound, so the perimeter is a product of many sides and very small side lengths. Even though this is an indeterminate product, the perimeter approaches the circle's circumference, which is finite and well defined.

Limits of Zero Times Infinity

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key terms and concepts

Learn how zero-times-infinity limits work, why they are indeterminate, and how rewriting the product as a quotient helps apply L'Hôpital's Rule.

Limits of Zero Times Infinity | JoVE K12

The JoVE Core Calculus collection provides a comprehensive visual guide to the entire calculus sequence, seamlessly connecting mathematical theory to real-world applications in physics and economics. Through dynamic animated lessons, the course transitions smoothly from foundational single-variable concepts to advanced multivariable topics. Students will explore everything from standard differentiation and integration to spatial vectors, multi-dimensional derivatives, and complex vector fields. By breaking down abstract coordinate systems, infinite series, and spatial motion, this structured approach builds the robust analytical skills and advanced mastery essential for STEM success.

calculus

Limits of Zero Times Infinity

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