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

Formas indeterminadas podem surgir em limites de produtos, onde um fator tende a zero e o outro tende a mais ou menos infinito.O resultado é indeterminado ou pouco claro. Pode ser zero, infinito ou um número finito.Por exemplo, considere o limite unilateral da seguinte função produto.À medida que x tende a zero, o primeiro fator x diminui para zero, enquanto o segundo fator, o logarítmico natural de um sobre x, tende para o infinito.O produto é algebricamente igual a um quociente que pode ser escrito em duas expressões diferentes.A primeira expressão cria uma forma infinita sobre infinito, enquanto a segunda expressão cria uma forma zero sobre zero.A Regra do L'Hôpital pode ser aplicada a qualquer um dos formulários. Mas, usar a primeira forma simplifica a diferenciação e resulta em zero.Uma visualização útil é um polígono regular inscrito em um círculo. À medida que o número de lados aumenta infinitamente, cada lado diminui em direção a zero.Isso cria um produto indeterminado: infinitos lados vezes comprimento zero. No entanto, o perímetro total se aproxima da circunferência fixa do círculo.Isso mostra como uma forma zero vezes infinita pode produzir um resultado significativo e finito.

produtos indeterminados

Indeterminate Products

Formas indeterminadas tamb&#xE9;m surgem na avalia&#xE7;&#xE3;o de limites envolvendo produtos, especialmente quando um fator tende a zero enquanto o outro tende ao infinito positivo ou negativo. Essa situa&#xE7;&#xE3;o, comumente descrita como uma forma do tipo zero vezes infinito, n&#xE3;o possui um resultado imediatamente interpret&#xE1;vel. Dependendo de como os fatores se comportam em rela&#xE7;&#xE3;o uns aos outros, o limite de tal produto pode ser zero, infinito ou um valor finito n&#xE3;o nulo.
Limites envolvendo produtos e reescrita alg&#xE9;brica
Para analisar limites desse tipo, o produto deve primeiro ser reescrito como um quociente para que a Regra de L&#x2019;H&#xF4;pital possa ser aplicada. Considere um limite lateral no qual a vari&#xE1;vel tende a zero, fazendo com que um fator diminua em dire&#xE7;&#xE3;o a zero enquanto o outro fator, como o logaritmo natural do inverso da vari&#xE1;vel, aumenta sem limites. Embora a express&#xE3;o original seja um produto, ela pode ser reescrita algebricamente em duas formas equivalentes de quociente. Uma reorganiza&#xE7;&#xE3;o produz uma forma PT_19834_1, enquanto a outra produz uma forma PT_19834_2. Ambas as representa&#xE7;&#xF5;es s&#xE3;o v&#xE1;lidas e permitem o uso da Regra de L&#x2019;H&#xF4;pital.
A aplica&#xE7;&#xE3;o da regra envolve derivar o numerador e o denominador separadamente e, em seguida, avaliar o limite resultante. Embora qualquer uma das formas do quociente possa ser usada, a escolha da forma pode afetar a simplicidade da deriva&#xE7;&#xE3;o. Neste caso, a forma de infinito sobre infinito resulta em uma derivada mais simples, revelando que o limite do produto original &#xE9; zero.
Interpreta&#xE7;&#xE3;o geom&#xE9;trica
Uma analogia geom&#xE9;trica &#xFA;til para entender formas indeterminadas do tipo zero vezes infinito &#xE9; um pol&#xED;gono regular inscrito em um c&#xED;rculo. &#xC0; medida que o n&#xFA;mero de lados do pol&#xED;gono aumenta sem limites, o comprimento de cada lado individual diminui em dire&#xE7;&#xE3;o a zero. O per&#xED;metro total, no entanto, &#xE9; o produto do n&#xFA;mero de lados pelo comprimento de cada lado. Isso produz um produto indeterminado envolvendo um n&#xFA;mero infinito de lados, cada um de comprimento que tende a zero. Apesar disso, o per&#xED;metro se aproxima de um valor finito e bem definido, igual &#xE0; circunfer&#xEA;ncia do c&#xED;rculo. Este exemplo ilustra como formas indeterminadas do tipo zero vezes infinito podem gerar limites finitos significativos quando analisadas adequadamente.

Produtos indeterminados

Formas indeterminadas tamb&#xE9;m surgem na avalia&#xE7;&#xE3;o de limites envolvendo produtos, especialmente quando um fator tende a zero enquanto o ...

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Calculus

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Applications of Differentiation

Aplicações da Diferenciação

KEY TERMS AND CONCEPTS

TERMOS E CONCEITOS-CHAVE

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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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...

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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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