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

Les limites décrivent le comportement d’une fonction lorsque l’entrée s’approche d’une valeur spécifique.Cela permet de comprendre le comportement de la fonction à proximité de points où elle n’est pas définie, comme un espace dans le graphique où une valeur est manquante, ou un saut discontinu où le graphique se déplace soudainement vers une autre valeur.Un exemple courant est une fonction par morceaux, qui change soudainement à une valeur d’entrée spécifique.Cela peut être visualisé à l’aide d’un interrupteur de thermostat intelligent qui maintient le chauffage éteint lorsque la température est supérieure à un niveau défini, mais s’allume instantanément lorsque la température descend en dessous. Sur le graphique, cela apparaît comme une sortie de 0 au-dessus de la température réglée et de 1 en dessous.Ce changement soudain crée des résultats différents selon la direction d’où provient l’entrée, un concept connu sous le nom de limite unilatérale.Lorsque l’entrée s’approche du niveau par la gauche ou le côté négatif, l’interrupteur reste éteint (c’est ce qu’on appelle la limite de gauche).Lorsque l’entrée s’approche de la droite ou du côté positif, le commutateur s’allume, c’est ce qu’on appelle la limite de droite.Étant donné que les limites de gauche et de droite ne sont pas égales, la limite globale à ce point n’existe pas.

types de limites

Les limites sont un concept math&#233;matique essentiel pour comprendre le comportement des fonctions lorsque la variable tend vers des valeurs sp&#233;cifiques, en particulier lorsque la fonction n&#8217;est pas d&#233;finie en ce point. Elles permettent de mettre en &#233;vidence des tendances et des discontinuit&#233;s en examinant les valeurs auxquelles une fonction tend plut&#244;t que sa valeur en ce point.
Les limites unilat&#233;rales portent sur la direction selon laquelle une valeur est approch&#233;e. Lorsqu&#8217;une fonction adopte des comportements distincts selon que la variable tend vers cette valeur par la gauche ou par la droite, les deux limites unilat&#233;rales peuvent ne pas co&#239;ncider. Dans ce cas, la limite bilat&#233;rale en ce point n&#8217;existe pas. Le comportement de type seuil des fonctions, o&#249; un changement brutal se produit une fois qu&#8217;une certaine valeur de la variable est atteinte, peut &#234;tre d&#233;crit efficacement &#224; l&#8217;aide de cette notion.
D&#8217;autres fonctions pr&#233;sentent, &#224; proximit&#233; d&#8217;une valeur particuli&#232;re de la variable, un comportement erratique ou oscillant. Aucune limite ne peut &#234;tre d&#233;finie lorsqu&#8217;une fonction varie rapidement sans se stabiliser autour d&#8217;une tendance pr&#233;cise. Ce comportement conduit g&#233;n&#233;ralement &#224; l&#8217;absence de limite, m&#234;me si la fonction est d&#233;finie au voisinage.
Les limites s&#8217;appliquent aux changements soudains, aux valeurs non d&#233;finies ou aux comportements impr&#233;visibles. Elles fournissent un cadre math&#233;matique pour analyser le comportement des fonctions autour de points critiques et d&#233;crire la nature du changement m&#234;me lorsqu&#8217;une &#233;valuation directe est impossible. Les applications incluent la mod&#233;lisation de la vitesse instantan&#233;e en physique et l&#8217;&#233;valuation des variations du co&#251;t marginal ou du profit en &#233;conomie.

Types de limites

Les limites sont un concept math&#233;matique essentiel pour comprendre le comportement des fonctions lorsque la variable tend vers des valeurs ...

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Math Fundamentals

Fondamentaux des mathématiques

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Limits

Limites

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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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Limits in Graphs and Real-World Change

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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

Vertical Asymptotes and Infinite Limits

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Oscillating Limits Near Discontinuities

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limit laws i

Limit Laws for Combining Functions

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limit laws ii

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Epsilon-Delta Logic for Limits

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limits at infinity

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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