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

Las formas indeterminadas pueden surgir en límites de productos, donde un factor tiende a cero y el otro tiende a más o menos infinito.El resultado es indeterminado o poco claro. Podría ser cero, infinito o un número finito.Por ejemplo, consideremos el límite unilateral de la siguiente función producto.A medida que x tiende a cero, el primer factor x disminuye a cero, mientras que el segundo factor, el logarítmic natural de uno sobre x, tiende a infinito.El producto es algebraicamente igual a un cociente que puede escribirse en dos expresiones diferentes.La primera expresión crea una forma infinita sobre infinito, mientras que la segunda expresión crea una forma cero sobre cero.La norma de L'Hôpital puede aplicarse a cualquiera de las dos formas. Pero, usar la primera forma simplifica la diferenciación y da cero como resultado.Una visualización útil es un polígono regular inscrito en un círculo. A medida que el número de lados aumenta infinitamente, cada lado se reduce hacia cero.Esto crea un producto indeterminado: infinitos lados por cero longitud. Sin embargo, el perímetro total se acerca a la circunferencia fija del círculo.Esto muestra cómo una forma cero por infinito puede producir un resultado significativo y finito.

productos indeterminados

Indeterminate Products

Las formas indeterminadas tambi&#xE9;n surgen al evaluar l&#xED;mites que involucran productos, particularmente cuando un factor tiende a cero mientras que el otro tiende a infinito positivo o negativo. Esta situaci&#xF3;n, com&#xFA;nmente descrita como una forma de cero por infinito, no tiene un resultado inmediatamente interpretable. Dependiendo del comportamiento relativo de los factores, el l&#xED;mite de dicho producto puede ser cero, infinito o un valor finito distinto de cero.
L&#xED;mites de producto y reescritura algebraica
Para analizar l&#xED;mites de este tipo, el producto debe reescribirse primero como un cociente para poder aplicar la regla de L&#x2019;H&#xF4;pital. Consideremos un l&#xED;mite unilateral en el que la variable tiende a cero, lo que hace que un factor tienda a cero mientras que el otro factor, como el logaritmo natural del rec&#xED;proco de la variable, crece sin l&#xED;mite. Aunque la expresi&#xF3;n original es un producto, puede reescribirse algebraicamente en dos formas de cociente algebraicamente equivalentes. Una reorganizaci&#xF3;n produce la forma PT_19834_1, mientras que la otra produce la forma PT_19834_2. Ambas representaciones son v&#xE1;lidas y permiten el uso de la regla de L&#x2019;H&#xF4;pital.
Aplicar la regla implica diferenciar el numerador y el denominador por separado y luego calcular el l&#xED;mite resultante. Si bien se puede usar cualquiera de las dos formas de cociente, la elecci&#xF3;n de la forma puede afectar la facilidad de la derivaci&#xF3;n. En este caso, la forma de infinito sobre infinito produce una derivada m&#xE1;s simple, lo que revela que el l&#xED;mite del producto original es cero.
Interpretaci&#xF3;n geom&#xE9;trica
Una analog&#xED;a geom&#xE9;trica &#xFA;til para comprender las formas indeterminadas de cero por infinito es un pol&#xED;gono regular inscrito en un c&#xED;rculo. A medida que el n&#xFA;mero de lados del pol&#xED;gono tiende a infinito, la longitud de cada lado disminuye hacia cero. Sin embargo, el per&#xED;metro total es el producto del n&#xFA;mero de lados por la longitud de cada lado. Esto produce un producto indeterminado que involucra una infinidad de lados de longitud nula. A pesar de esto, el per&#xED;metro se aproxima a un valor finito y bien definido, igual a la circunferencia del c&#xED;rculo. Este ejemplo ilustra c&#xF3;mo las formas de cero por infinito pueden producir l&#xED;mites finitos significativos cuando se analizan adecuadamente.

Productos indeterminados

Las formas indeterminadas tambi&#xE9;n surgen al evaluar l&#xED;mites que involucran productos, particularmente cuando un factor tiende a cero mientras ...

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Calculus

Cálculo

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Unidad

Applications of Differentiation

Aplicaciones de la derivación

KEY TERMS AND CONCEPTS

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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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