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

Ein unendlicher Grenzwert wird in einem Diagramm angezeigt, wenn die Ausgabe einer Funktion unbegrenzt zu- oder abnimmt, wenn sich die Eingabe einem bestimmten Wert nähert.Mathematisch bedeutet dies, dass die Grenze von f(x) gegen unendlich zunimmt oder gegen negative Unendlichkeit abnimmt, wenn x sich a nähert.Dieses Verhalten weist auf das Vorhandensein einer vertikalen Asymptote hin, d. h. einer vertikalen Linie, der sich die Funktion nähert, die sie jedoch nie berührt oder kreuzt.Wenn x z. B. gleich 2 ist, ist die Funktion nicht definiert. Aber wenn x sich 2 von beiden Seiten nähert, steigen die Werte der Funktion schnell und unbegrenzt an. Im Diagramm erscheint dies als steiler Anstieg, der eine vertikale Asymptote bei x = 2 bildet.Eine hilfreiche Analogie ist ein Kletterer, der sich einer Klippe nähert. Der Weg beginnt mit einem leichten Gefälle, aber je weiter der Kletterer kommt, desto steiler wird die Steigung.In der Nähe der Kante dreht sich der Hang fast senkrecht – zu steil, um weiterzugehen.Der Graph dieses Szenarios zeigt eine Kurve, die immer steiler ansteigt und widerspiegelt, wie die Werte einer Funktion in der Nähe einer vertikalen Asymptote unkontrollierbar ansteigen.

arten von grenzwerten ii

Beobachtet man das Verhalten einer Kurve in der N&#228;he eines bestimmten Punktes entlang der horizontalen Achse, so gibt es F&#228;lle, in denen die H&#246;he der Kurve mit zunehmender Ann&#228;herung an diesen Punkt unbegrenzt zu- oder abnimmt. Die Kurve pendelt sich auf keinem bestimmten Wert ein; vielmehr werden die Werte &#8211; nach oben oder unten &#8211; umso extremer, je n&#228;her sie dem Punkt kommen. An der betreffenden Stelle existiert kein wohldefinierter Funktionswert, dennoch wird das umgebende Verhalten immer ausgepr&#228;gter und deutet auf einen abrupten Richtungswechsel hin.
Die Werte k&#246;nnen auf einer Seite des Punktes stark ansteigen und auf der anderen ebenso stark abfallen, was einen auff&#228;lligen Kontrast erzeugt. Der Graph zeigt in einem schmalen Bereich eine steile Spitze oder eine tiefe Senke, in dem sie nahezu senkrecht nach oben oder unten strebt. Durch den betreffenden Punkt verl&#228;uft keine stetige Fortsetzung. Stattdessen durchbricht die Kurve ihr &#252;bliches Muster und wird auf sehr kurzer horizontaler Strecke nahezu vertikal.
Stellen Sie sich eine Person vor, die einen Weg entlang geht, der zun&#228;chst sanft ansteigt, aber mit jedem Schritt steiler wird. Die Anstrengung beim Aufstieg nimmt rasch zu, bis der Weg vor ihr nahezu senkrecht wird und ein weiteres Vorankommen unm&#246;glich macht. Die Form des Graphen spiegelt dasselbe Muster wider. Wenn sich die Position dem kritischen Punkt n&#228;hert, &#228;ndern sich die Werte so schnell, dass der Graph seine glatte Gestalt aufgibt und unkontrolliert ansteigt oder abf&#228;llt &#8211; wie der Pfad am Rand einer Klippe.

Arten von Grenzwerten II

Beobachtet man das Verhalten einer Kurve in der N&#228;he eines bestimmten Punktes entlang der horizontalen Achse, so gibt es F&#228;lle, in denen die ...

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ZENTRALE BEGRIFFE UND KONZEPTE

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

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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Finding instantaneous velocity from motion starts with a simple case: a vehicle moving along a straight, level path. Its position can be tracked at any time. The first step is to measure how far the vehicle travels over a fixed time period. That measurement is average velocity. It is found by dividing the total change in position by the time interval. Average velocity gives a broad picture of how quickly the vehicle moves between two times. It does not show how the motion changes at one exact...

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 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...

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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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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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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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limits at infinity

Horizontal Asymptotes in Function Graphs

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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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