Name: oscillating limits near discontinuities
Uploaded: 2025-11-26T05:43:41.000Z
Duration: 1 min 19 s
Channel: JoVE

Considere una rueda de bicicleta con radios muy espaciados que giran muy rápido.En el borde exterior, los radios se mueven lo suficientemente lento como para permanecer visibles individualmente. Más cerca del buje, los radios pasan más rápidamente. En el centro, el movimiento es demasiado rápido para que el ojo humano lo distinga, y los radios aparecen borrosos.Esta rueca proporciona una analogía física para el concepto de una discontinuidad oscilante.Se puede estudiar usando la ecuación, el límite de la función seno uno sobre x ya que x tiende a cero.Al igual que un punto en una rueda giratoria nunca se asienta en una posición fija, la función no puede converger a un valor específico.A medida que la entrada x se acerca a cero, uno sobre x crece hacia el infinito positivo o negativo.Debido a que la función seno circula repetidamente entre –1 y +1, la salida de seno uno sobre x oscila entre estos dos valores infinitamente. Las oscilaciones se vuelven cada vez más frecuentes a medida que x se acerca a cero.Dado que la salida nunca se establece en un valor, el límite en cero no existe.

límites con discontinuidades oscilantes

Una discontinuidad oscilante es un tipo de discontinuidad en la que los valores de una funci&#243;n fluct&#250;an de manera infinita a medida que la variable de entrada se aproxima a un punto determinado. A diferencia de las discontinuidades de salto, donde la funci&#243;n cambia repentinamente entre dos valores, o de las discontinuidades infinitas, donde la funci&#243;n diverge sin estar acotada, una discontinuidad oscilante surge de una r&#225;pida variaci&#243;n de vaiv&#233;n. Dado que la funci&#243;n nunca se estabiliza hacia un &#250;nico valor, no existe un l&#237;mite finito en ese punto.Uno de los ejemplos m&#225;s conocidos es la funci&#243;n<figure class='image image-style-align-center image_resized' style='display:table;margin-left:auto;margin-right:auto;width:90.21%;'><img style='aspect-ratio:300/20;' src='https://cloudfront.jove.com/files/ftp_upload/41305/41305_Equation_1_editor_41305.svg' alt='41305_Equation_1.svg' /></figure>A medida que x tiende a cero, el t&#233;rmino rec&#237;proco 1/x crece sin estar acotado. La funci&#243;n seno, al ser peri&#243;dica y definida entre &#8722;1 y 1, hace que los valores de la funci&#243;n oscilen indefinidamente dentro de esos l&#237;mites. Cuanto m&#225;s se acerca x a cero, m&#225;s r&#225;pidamente se producen las oscilaciones, lo que origina una densidad infinita de fluctuaciones cerca del origen. A pesar de estar acotada, la funci&#243;n no tiene un l&#237;mite bien definido en cero.Este fen&#243;meno puede visualizarse mediante analog&#237;as f&#237;sicas. Por ejemplo, una rueda de bicicleta giratoria con radios muy juntos parece pasar m&#225;s r&#225;pido cuanto m&#225;s se mira cerca del eje. Finalmente, el movimiento se vuelve tan r&#225;pido que los radios se confunden visualmente. De igual forma, en el caso matem&#225;tico, las oscilaciones de la funci&#243;n se comprimen tanto cerca de cero que la funci&#243;n nunca se estabiliza en un &#250;nico valor.Las discontinuidades oscilantes ilustran la sutil diferencia entre acotaci&#243;n y convergencia en el an&#225;lisis. Demuestran c&#243;mo una funci&#243;n puede permanecer dentro de l&#237;mites num&#233;ricos fijos y, aun as&#237;, no poseer un l&#237;mite, lo que pone de relieve la precisi&#243;n requerida en el c&#225;lculo al analizar la continuidad y el comportamiento de las funciones cerca de puntos singulares.

Límites con discontinuidades oscilantes

Una discontinuidad oscilante es un tipo de discontinuidad en la que los valores de una funci&#243;n fluct&#250;an de manera infinita a medida que la ...

Education

Educación

JoVE Core

Math Fundamentals

Fundamentos de matemáticas

Unit

Unidad

Limits

Límites

KEY TERMS AND CONCEPTS

TÉRMINOS Y CONCEPTOS CLAVE

Limits with Oscillating Discontinuities

oscillating limits near discontinuities

Oscillating discontinuities happen when a function keeps changing value as the input gets close to a point. The output does not settle down to one number. Because of that, the limit at that point does not exist.

This kind of discontinuity is different from a jump discontinuity and an infinite discontinuity. In a jump discontinuity, the function shifts suddenly between two values. In an infinite discontinuity, the function grows without bound.

A well-known example comes from the function involving sine and a reciprocal term near x = 0. As x approaches zero, the reciprocal term grows without bound. At the same time, the sine function stays between -1 and 1, so the output keeps oscillating between those bounds.

The closer x gets to zero, the faster the oscillations happen. The fluctuations become extremely dense near the origin. Even though the function stays bounded, it still has no well-defined limit at zero.

A physical picture can help explain this behavior. A spinning bicycle wheel with closely spaced spokes seems to flash faster as you look closer to the hub. Soon the motion becomes so fast that the spokes blur together.

The math works in a similar way. Near zero, the function changes too quickly to settle on a single value. Oscillating discontinuities show that a function can stay within fixed limits and still fail to converge, which is an important idea in calculus.

Oscillating Limits Near Discontinuities

tangent to a curve

Secant Lines Approaching a Tangent

Secant lines and tangent lines help describe how a curve changes at a chosen point. The curve shown here comes from a function where each output is the square of the input. It bends upward and becomes steeper as you move farther from the center. To study the curve near one position, a second point is chosen slightly ahead on the graph. A straight line through the two points is called a secant line. It crosses the curve and shows the average rise or fall between those points. Because it spans a...

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introduction to limits

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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...

types of limits ii

Vertical Asymptotes and Infinite Limits

Vertical asymptotes and infinite limits describe how a curve behaves near a point where its height grows without bound. As the position on the horizontal axis gets closer to that point, the graph does not settle at a single value. Instead, the values rise or fall more and more sharply. At the point itself, no defined value exists. The curve may climb steeply on one side and drop just as sharply on the other. This creates a strong contrast near the same x-value. On the graph, this behavior...

limits with oscillating discontinuities

Oscillating discontinuities happen when a function keeps changing value as the input gets close to a point. The output does not settle down to one number. Because of that, the limit at that point does not exist. This kind of discontinuity is different from a jump discontinuity and an infinite discontinuity. In a jump discontinuity, the function shifts suddenly between two values. In an infinite discontinuity, the function grows without bound. A well-known example comes from the function...

limit laws i

Limit Laws for Combining Functions

Limit laws help students find limits of combined functions. These rules show how a function behaves as its input approaches a specific value. They are useful when the limit of each separate function exists. The Sum and Difference Laws say that the limit of a sum or difference equals the sum or difference of the individual limits. The Product Law says that the limit of a product equals the product of the separate limits. These rules make many limit problems easier to solve. A car rental cost...

limit laws ii

Quotient, Power, and Root Limits

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limits at infinity

Horizontal Asymptotes in Function Graphs

Horizontal asymptotes describe how a function behaves as the input grows very large. In this topic, a decreasing function gets smaller and smaller as the input increases. Its output moves closer and closer to one fixed value. The function does not reach that value, but it approaches it without limit. On a graph, this long-term behavior appears as a curve that flattens out. The graph seems to level off near a horizontal line. That line is the horizontal asymptote, which the curve approaches but...

key terms and concepts

Explore oscillating discontinuities, including how rapid sine-based fluctuations near zero prevent a limit from existing, with a bicycle wheel analogy.

Oscillating Limits Near Discontinuities | 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

Oscillating Limits Near Discontinuities

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