Name: horizontal asymptotes in function graphs
Uploaded: 2025-11-26T05:46:03.000Z
Duration: 1 min 24 s
Channel: JoVE

Les limites d’une fonction peuvent être évaluées lorsque x tend vers l’infini positif ou négatif. Ces deux limites sont distinctes et doivent être vérifiées séparément.Considérons la fonction x au cube. Lorsque x tend vers l’infini positif, la valeur augmente sans borne.Lorsque x tend vers moins l’infini, la valeur diminue sans borne.En revanche, la fonction sinus oscille entre −1 et 1. Comme il ne s’installe jamais, sa limite à l’infini n’existe pas.Certaines fonctions s’approchent d’une valeur finie, telle que une divisée par x plus 2. Comme x tend vers l’infini, un sur x devient zéro, laissant la valeur 2. Cette droite horizontale, y égale 2, est appelée asymptote horizontale.Ce concept apparaît dans des circuits réels, par exemple lorsqu’un condensateur est chargé dans un circuit RC en série.Lorsqu’une batterie est connectée, la charge du condensateur augmente avec le temps. En prenant la limite lorsque le temps t tend vers l’infini, le terme exponentiel sera zéro, et la charge du condensateur s’approchera d’une valeur maximale constante, qui représente l’asymptote horizontale de la courbe.

limites à l’infini

La fonction qui d&#233;cro&#238;t lorsque la variable devient tr&#232;s grande offre un exemple clair de la mani&#232;re dont une fonction peut se comporter pour des valeurs extr&#234;mes. Lorsque la variable augmente contin&#251;ment, la valeur de la fonction devient de plus en plus petite, se rapprochant d&#8217;une valeur fixe. Bien que la fonction n&#8217;atteigne jamais effectivement cette valeur, elle s&#8217;en approche ind&#233;finiment. Ce comportement constitue un concept fondamental pour comprendre la fa&#231;on dont une fonction se comporte lorsque la variable cro&#238;t sans borne. La repr&#233;sentation graphique de cette fonction montre que sa courbe s&#8217;aplatit et semble se stabiliser &#224; proximit&#233; d&#8217;une droite horizontale donn&#233;e. Cette droite, que la courbe approche sans jamais la toucher, repr&#233;sente le comportement asymptotique de la fonction.
Un autre exemple de ce type de comportement appara&#238;t pour une fonction qui se stabilise progressivement &#224; des valeurs diff&#233;rentes selon que la variable augmente ou diminue sans borne. La valeur de la fonction se rapproche d&#8217;une certaine valeur limite lorsque la variable tend vers +&#8734;. Inversement, lorsque la variable tend vers &#8722;&#8734;, la valeur de la fonction se rapproche d&#8217;une autre valeur, inf&#233;rieure. Ces deux bornes ne sont ni franchies ni atteintes par la fonction, mais elles indiquent la mani&#232;re dont celle-ci se comporte lorsque la variable tend vers des valeurs extr&#234;mes, dans un sens comme dans l&#8217;autre. Sur un graphique, la courbe de la fonction s&#8217;incurve doucement et s&#8217;approche de ces limites sup&#233;rieure et inf&#233;rieure, illustrant une transition continue, sans changements brusques.
Ce type d&#8217;analyse aide &#224; comprendre le comportement des fonctions &#224; long terme et s&#8217;av&#232;re essentiel dans de nombreux domaines des math&#233;matiques, notamment pour d&#233;crire des tendances, estimer des valeurs et mod&#233;liser des ph&#233;nom&#232;nes r&#233;els de mani&#232;re abstraite. Ces caract&#233;ristiques sont d&#233;terminantes pour analyser la stabilit&#233; ou les limites de syst&#232;mes repr&#233;sent&#233;s par des fonctions math&#233;matiques.

Limites à l’infini

La fonction qui d&#233;cro&#238;t lorsque la variable devient tr&#232;s grande offre un exemple clair de la mani&#232;re dont une fonction peut se ...

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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 never touches. This helps show the end behavior of the function.

The same idea also appears in a function with two different boundaries. As the input becomes very large in the positive direction, the output approaches an upper value. As the input becomes very large in the negative direction, the output approaches a different lower value. These two values are not crossed or reached by the function.

A graph of this function bends smoothly toward both boundaries. It changes gradually without sudden jumps. Studying these limits at extreme input values helps explain long-run trends. It is useful for describing patterns, estimating values, and modeling real-world phenomena in abstract form. This kind of analysis also helps show the stability of systems represented by functions.

Horizontal Asymptotes in Function Graphs

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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introduction to limits

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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

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

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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...

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limits at infinity

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

Learn how function graphs level off at horizontal asymptotes, including one-sided behavior as inputs grow large in positive or negative directions.

Horizontal Asymptotes in Function Graphs | 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.

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Horizontal Asymptotes in Function Graphs

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