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

Die Grenzen einer Funktion können ausgewertet werden, wenn x sich der positiven oder negativen Unendlichkeit nähert. Diese beiden Grenzwerte sind unterschiedlich und müssen separat überprüft werden.Betrachten Sie die Funktion x cubed. Wenn x sich positiver Unendlichkeit nähert, steigt der Wert unbegrenzt an.Wenn x sich dem negativen Unendlichen nähert, nimmt der Wert unbegrenzt ab.Im Gegensatz dazu oszilliert die Sinusfunktion zwischen -1 und 1. Da es sich nie absetzt, existiert seine Grenze im Unendlichen nicht.Einige Funktionen nähern sich einem endlichen Wert, z. B. eins geteilt durch x plus 2. Da x gegen unendlich tendiert, wird eins über x zu Null, so dass der Wert 2 übrig bleibt. Diese horizontale Linie, y ist gleich 2, wird als horizontale Asymptote bezeichnet.Dieses Konzept tritt in realen Schaltungen auf, z. B. wenn ein Kondensator in einer Reihen-RC-Schaltung geladen wird.Wenn eine Batterie angeschlossen ist, steigt die Ladung des Kondensators mit der Zeit an. Nimmt man den Grenzwert, wenn sich die Zeit t unendlich nähert, so ist der Exponentialterm Null, und die Ladung des Kondensators nähert sich einem konstanten Maximalwert, der die horizontale Asymptote der Kurve darstellt.

grenzen im unendlichen

Eine Funktion, deren Ausgabewerte mit wachsendem Eingabewert abnehmen, liefert ein anschauliches Beispiel f&#252;r das Verhalten mathematischer Funktionen bei extremen Eingabewerten. Steigt die Eingabe kontinuierlich, werden die Funktionswerte immer kleiner und n&#228;hern sich einem festen Wert. Obwohl dieser Wert nie tats&#228;chlich erreicht wird, n&#228;hert er sich diesem Wert beliebig an. Dieses Verhalten ist grundlegend f&#252;r das Verst&#228;ndnis des Funktionsverhaltens bei gegen unendlich wachsender Eingabe. In der grafischen Darstellung flacht die Kurve ab und scheint sich in der N&#228;he einer bestimmten horizontalen Geraden zu stabilisieren. Diese Linie, der sich der Graph n&#228;hert, ohne sie zu ber&#252;hren, beschreibt das langfristige Verhalten der Funktion.
Ein weiteres Beispiel zeigt eine Funktion, die sich abh&#228;ngig von der Richtung des unbegrenzten Wachstums des Eingabewerts bei unterschiedlichen Werten ann&#228;hert. F&#252;r x &#8594; +&#8734; n&#228;hert sich der Funktionswert einem oberen Grenzwert, f&#252;r x &#8594; &#8722;&#8734; einem anderen unteren Grenzwert. Diese beiden Schranken werden von der Funktion weder &#252;berschritten noch erreicht; sie kennzeichnen vielmehr das Verhalten der Funktion, wenn sich die Eingabe in beide Richtungen Extremwerten n&#228;hert. Im Diagramm biegt sich die Kurve sanft und n&#228;hert sich diesen oberen und unteren Schranken an; der &#220;bergang ist glatt und ohne sprunghafte &#196;nderungen.
Diese Art der Analyse unterst&#252;tzt das Verst&#228;ndnis des Langzeit- bzw. asymptotischen Verhaltens von Funktionen und ist in vielen Bereichen der Mathematik wesentlich &#8211; insbesondere zur Beschreibung von Trends, zur Absch&#228;tzung von Werten und zur abstrakten Modellierung realer Ph&#228;nomene. Diese Eigenschaften sind bedeutsam, wenn Stabilit&#228;t oder Grenzverhalten von durch Funktionen dargestellten Systemen untersucht werden.

Grenzen im Unendlichen

Eine Funktion, deren Ausgabewerte mit wachsendem Eingabewert abnehmen, liefert ein anschauliches Beispiel f&#252;r das Verhalten mathematischer Funktionen ...

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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...

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Oscillating Limits Near Discontinuities

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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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