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

Prenons l’exemple d’une roue de vélo avec des rayons rapprochés qui tourne très vite.Sur le bord extérieur, les rayons se déplacent suffisamment lentement pour rester visibles individuellement. Plus près du moyeu, les rayons défilent plus rapidement. Au centre, le mouvement est trop rapide pour que l’œil humain puisse le distinguer et les rayons apparaissent flous.Ce rouet fournit une analogie physique pour le concept de discontinuité oscillante.Elle peut être étudiée à l’aide de l’équation, la limite de la fonction sinus un sur x comme x tend vers zéro.Tout comme un point sur une roue tournante ne s’installe jamais dans une position fixe, la fonction ne parvient pas à converger vers une valeur spécifique.Lorsque l’entrée x tend vers zéro, un sur x croît vers l’infini positif ou négatif.Étant donné que la fonction sinus effectue des cycles répétés entre –1 et +1, la sortie du sinus un sur x oscille entre ces deux valeurs à l’infini. Les oscillations deviennent de plus en plus fréquentes à mesure que x se rapproche de zéro.Comme la sortie ne se fixe jamais sur une seule valeur, la limite à zéro n’existe pas.

limites en présence de discontinuités oscillantes

Une discontinuit&#233; oscillante est un type de discontinuit&#233; o&#249; les valeurs d&#8217;une fonction fluctuent ind&#233;finiment lorsque la variable tend vers un point donn&#233;. Contrairement aux discontinuit&#233;s par saut, o&#249; la fonction bascule brusquement entre deux valeurs, ou aux discontinuit&#233;s infinies, o&#249; la fonction diverge vers l&#8217;infini, une discontinuit&#233; oscillante r&#233;sulte d&#8217;une variation rapide en va-et-vient. Comme la fonction ne tend vers aucune valeur unique, aucune limite finie n&#8217;existe en ce point.L&#8217;un des exemples les plus connus est la fonction<figure class='image image-style-align-center image_resized' style='display:table;margin-left:auto;margin-right:auto;width:100%;'><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>Lorsque x tend vers z&#233;ro, l&#8217;argument du sinus, 1/x, devient arbitrairement grand. La fonction sinus, p&#233;riodique et born&#233;e entre &#8722;1 et 1, contraint alors la valeur de la fonction &#224; osciller sans fin entre ces bornes. &#192; mesure que x tend vers z&#233;ro, les oscillations se densifient, ce qui se traduit par une densit&#233; de fluctuations tendant vers l&#8217;infini au voisinage de l&#8217;origine. Bien que born&#233;e, la fonction ne poss&#232;de pas de limite en 0.Ce ph&#233;nom&#232;ne peut &#234;tre visualis&#233; &#224; l&#8217;aide d&#8217;analogies physiques. Par exemple, une roue de bicyclette en rotation dont les rayons sont tr&#232;s rapproch&#233;s semble d&#233;filer de plus en plus vite lorsqu&#8217;on regarde pr&#232;s du moyeu. Finalement, le mouvement devient si rapide que les rayons se brouillent. De m&#234;me, dans le cas math&#233;matique, les oscillations de la fonction se compriment &#224; l&#8217;approche de z&#233;ro, &#224; tel point que la fonction ne se fixe jamais sur une valeur unique.Les discontinuit&#233;s oscillantes mettent en &#233;vidence la diff&#233;rence subtile entre le caract&#232;re born&#233; et la convergence en analyse. Elles montrent comment une fonction peut rester dans des limites num&#233;riques fixes tout en ne poss&#233;dant aucune limite, ce qui souligne la pr&#233;cision requise en analyse pour d&#233;crire la continuit&#233; et le comportement des fonctions au voisinage de points singuliers.

Limites en présence de discontinuités oscillantes

Une discontinuit&#233; oscillante est un type de discontinuit&#233; o&#249; les valeurs d&#8217;une fonction fluctuent ind&#233;finiment lorsque la ...

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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

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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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Limits describe the value a function approaches as its input gets closer to a point. They are useful even when the function is undefined at that exact value. This makes limits a key idea in calculus and an important step toward understanding continuity, derivatives, and integrals. In math, a function f(x) has a limit L at x = a when its output approaches L as x gets very close to a. This idea is written with limit notation, and it does not require f(a) to be defined. The notation shows the...

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One-Sided Limits and Limit Failure

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Vertical Asymptotes and Infinite Limits

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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

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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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