Name: vertical asymptotes and infinite limits
Uploaded: 2025-11-26T05:43:42.000Z
Duration: 1 min 24 s
Channel: JoVE

Une limite infinie apparaît sur un graphique lorsque la sortie d’une fonction augmente ou diminue sans limite lorsque l’entrée s’approche d’une valeur spécifique.Mathématiquement, cela signifie que la limite de f(x) augmente vers l’infini ou diminue vers l’infini négatif lorsque x tend vers a.Ce comportement indique la présence d’une asymptote verticale, c’est-à-dire une ligne verticale que la fonction approche, mais ne touche ni ne franchit jamais.Par exemple, lorsque x est égal à 2, la fonction est indéfinie. Mais lorsque x s’approche de 2 de chaque côté, les valeurs de la fonction augmentent rapidement sans borne. Sur le graphique, cela apparaît comme une montée abrupte, formant une asymptote verticale en x = 2.Une analogie utile est celle d’un grimpeur qui s’approche d’une falaise. Le chemin commence par une pente douce, mais au fur et à mesure que le grimpeur avance, la pente s’accentue rapidement.Près du bord, la pente devient presque verticale, trop raide pour continuer.Le graphique de ce scénario révèle une courbe qui grimpe de plus en plus abruptement, reflétant comment, près d’une asymptote verticale, les valeurs d’une fonction augmentent au-delà de tout contrôle.

types de limites ii

Lorsqu&#8217;on observe le comportement d&#8217;une courbe au voisinage d&#8217;un point donn&#233; sur l&#8217;axe horizontal, il existe des cas o&#249; son ordonn&#233;e augmente ou diminue sans borne &#224; mesure que l&#8217;on s&#8217;en approche. La courbe ne se stabilise &#224; aucune valeur particuli&#232;re&#160;; au contraire, les valeurs deviennent de plus en plus extr&#234;mes &#8212; vers le haut ou vers le bas &#8212; &#224; mesure qu&#8217;elles s&#8217;en rapprochent. Aucune valeur d&#233;finie n&#8217;est atteinte exactement en ce point, mais le comportement au voisinage s&#8217;amplifie, signalant une variation brusque de direction.
Les valeurs peuvent cro&#238;tre fortement d&#8217;un c&#244;t&#233; du point et chuter tout aussi brusquement de l&#8217;autre, cr&#233;ant un contraste net. Le graphique pr&#233;sente alors une pointe ou un creux tr&#232;s marqu&#233; dans une r&#233;gion &#233;troite, o&#249; la courbe se dresse presque &#224; la verticale. Aucun prolongement lisse ne passe par le point consid&#233;r&#233;. Au contraire, le trac&#233; rompt avec son comportement habituel et devient quasi vertical sur un tr&#232;s court intervalle horizontal.
Imaginez une personne marchant sur un sentier qui commence par une pente douce mais devient de plus en plus raide &#224; chaque pas. L&#8217;effort &#224; fournir cro&#238;t rapidement jusqu&#8217;&#224; ce que le chemin qui s&#8217;annonce soit presque vertical, rendant toute progression impossible. Le trac&#233; du graphique refl&#232;te le m&#234;me ph&#233;nom&#232;ne. &#192; mesure que la position se rapproche du point critique, les valeurs varient si rapidement que le graphique perd sa r&#233;gularit&#233;, s&#8217;&#233;levant ou s&#8217;affaissant de fa&#231;on incontr&#244;l&#233;e, comme le sentier au bord d&#8217;une falaise.

Types de limites II

Lorsqu&#8217;on observe le comportement d&#8217;une courbe au voisinage d&#8217;un point donn&#233; sur l&#8217;axe horizontal, il existe des cas o&#249; ...

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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 appears as a narrow spike or dip. The curve stretches almost straight up or down in a very short horizontal space. No smooth line passes through the point, so the graph breaks its usual shape.

A simple way to picture this is a trail that starts with a gentle slope and becomes steeper with each step. The effort keeps increasing until the path is almost vertical. The graph shows the same kind of change, where the values near the point rise or fall uncontrollably, like reaching the edge of a cliff.

Vertical Asymptotes and Infinite Limits

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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...

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...

types of limits ii

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 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...

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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Quotient, Power, and Root Limits

Limit laws help students break down complicated expressions in calculus. They make it easier to find a limit by working with smaller parts of a function. In this lesson, the focus is on quotients, powers, and roots. The Quotient Law applies to a division of two functions. It says the limit of the quotient can be found by dividing the limits of the top and bottom functions, as long as the limit of the denominator exists and is not zero. This rule is useful when a limit problem contains a...

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Using continuity to find limits is a simple way to evaluate a function near a specific input value. This method works for algebraic expressions that have no breaks, holes, or jumps near the point of interest. When a function is continuous at that point, the limit can often be found by direct substitution. Direct substitution means replacing the variable with the value it approaches. The result gives the output value as the input nears that point. In a continuous function, the value near the...

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Using bounds to find a limit is the main idea behind the Squeeze Theorem. It helps when a function changes in a way that makes direct limit evaluation difficult. Some functions may oscillate quickly or follow irregular patterns near a specific input value. The theorem works by placing the function between two other functions near that point. The function must stay greater than or equal to a lower bound and less than or equal to an upper bound. If both outer functions approach the same limit as...

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Epsilon-Delta Logic for Limits

Epsilon-delta logic gives a precise way to describe a limit in calculus. It shows how a function behaves near a chosen point without using vague words like "getting close." This makes limit analysis clear and logically consistent. The formal definition says that the limit of f(x) as x approaches a is L. In symbols, for every epsilon greater than 0, there exists a delta greater than 0 such that the function values stay within epsilon of L whenever x stays within delta of a. In other words, no...

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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

Learn how graphs behave near vertical asymptotes and infinite limits, where values rise or fall without bound and the curve becomes nearly vertical.

Vertical Asymptotes and Infinite Limits | 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

Vertical Asymptotes and Infinite Limits

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