Name: one-sided limits and limit failure
Uploaded: 2025-11-26T05:43:43.000Z
Duration: 1 min 23 s
Channel: JoVE

Los límites describen cómo se comporta una función a medida que la entrada se acerca a un valor específico.Esto ayuda a comprender el comportamiento de la función cerca de puntos donde no está definida, como un espacio en el gráfico donde falta un valor o un salto discontinuo donde el gráfico cambia repentinamente a otro valor.Un ejemplo común es una función por partes, que cambia repentinamente en un valor de entrada específico.Esto se puede visualizar usando un interruptor de termostato inteligente que mantiene el calentador apagado cuando la temperatura está por encima de un nivel establecido, pero se enciende instantáneamente una vez que la temperatura desciende por debajo de él. En el gráfico, esto aparece como una salida de 0 por encima de la temperatura establecida y 1 por debajo de ella.Este cambio repentino crea diferentes resultados dependiendo de la dirección desde la que se acerca la entrada, un concepto conocido como límite unilateral.Cuando la entrada se acerca al nivel desde el lado izquierdo o negativo, el interruptor permanece apagado, esto se denomina límite de la izquierda.Cuando la entrada se acerca desde el lado derecho o positivo, el interruptor se enciende, esto se denomina límite de la derecha.Dado que los límites de la mano izquierda y la mano derecha no son iguales, el límite general en ese punto no existe.

tipos de límites

Los l&#237;mites son un concepto matem&#225;tico clave para comprender el comportamiento de las funciones cuando su variable de entrada se aproxima a valores espec&#237;ficos, especialmente cuando la funci&#243;n no est&#225; definida. Ayudan a revelar tendencias y discontinuidades al examinar los valores a los que se aproxima una funci&#243;n en lugar de su valor real.
Los l&#237;mites unilaterales se centran en la direcci&#243;n desde la que la variable se aproxima a un valor. Cuando una funci&#243;n se comporta de forma diferente seg&#250;n que la variable se aproxime desde la izquierda o desde la derecha, los dos l&#237;mites unilaterales pueden no coincidir. En tal caso, el l&#237;mite general en ese punto no existe. El comportamiento umbral en ciertas funciones, donde se produce un cambio abrupto cuando se alcanza un determinado valor de la variable, puede describirse eficazmente mediante esta idea.
Otras funciones presentan un comportamiento err&#225;tico u oscilante cerca de un valor espec&#237;fico de la variable. No puede establecerse un l&#237;mite consistente cuando la funci&#243;n var&#237;a con rapidez sin asentarse en una tendencia concreta. Este comportamiento suele implicar que el l&#237;mite no existe, aunque la funci&#243;n est&#233; definida en las proximidades.
Los l&#237;mites pueden aplicarse a cambios repentinos, valores indefinidos o patrones impredecibles. Proporcionan un marco matem&#225;tico para analizar el comportamiento de las funciones en el entorno de puntos cr&#237;ticos y describir la naturaleza del cambio, incluso cuando la evaluaci&#243;n directa es imposible. Sus aplicaciones incluyen el modelado de la velocidad instant&#225;nea en f&#237;sica y la evaluaci&#243;n de cambios en el coste marginal o en el beneficio econ&#243;mico.

Tipos de límites

Los l&#237;mites son un concepto matem&#225;tico clave para comprender el comportamiento de las funciones cuando su variable de entrada se aproxima a ...

Education

Educación

JoVE Core

Math Fundamentals

Fundamentos de matemáticas

Unit

Unidad

Limits

Límites

KEY TERMS AND CONCEPTS

TÉRMINOS Y CONCEPTOS CLAVE

Types of Limits I

one-sided limits and limit failure

Limits describe how a function behaves as its input gets close to a target value. They are useful when the function is undefined at that point. Instead of focusing on the exact value, limits show the trend near a critical point and can reveal a discontinuity.

One-sided limits look at approach from a single direction. A left-hand limit checks values coming from the left, while a right-hand limit checks values coming from the right. If a function changes differently from each side, the two one-sided limits may not match. In that case, the full limit at that point does not exist.

Some functions also show threshold-like behavior. That means the output changes abruptly once the input reaches a certain value. Limits help describe that sudden shift. Other functions can behave erratically or oscillate near a point. When the values do not settle into one clear trend, no consistent limit can be found, even if the function is defined nearby.

Limits are useful for studying sudden changes, undefined values, and unpredictable patterns. They give a way to analyze function behavior around important points even when direct evaluation is not possible. This idea is also used in physics to model instantaneous velocity and in economics to study marginal cost or profit changes.

One-Sided Limits and Limit Failure

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types of limits i

Limits describe how a function behaves as its input gets close to a target value. They are useful when the function is undefined at that point. Instead of focusing on the exact value, limits show the trend near a critical point and can reveal a discontinuity. One-sided limits look at approach from a single direction. A left-hand limit checks values coming from the left, while a right-hand limit checks values coming from the right. If a function changes differently from each side, the two...

types of limits ii

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limit laws i

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Horizontal Asymptotes in Function Graphs

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key terms and concepts

Learn how one-sided limits work, why a limit may not exist, and how limits describe sudden changes, oscillations, and real-world rates.

One-Sided Limits and Limit Failure | JoVE K12

The JoVE Math Fundamentals collection is designed to build a deep and comprehensive understanding of essential mathematics. Covering the foundations of algebra, geometry, trigonometry, and analytic reasoning, this course establishes a strong base for future studies. The collection connects core mathematical concepts with real-world practice, equipping learners to solve complex problems, master problem-solving skills, and apply abstract ideas.

math fundamentals

One-Sided Limits and Limit Failure

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