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

Los límites de una función se pueden evaluar a medida que x se acerca al infinito positivo o negativo. Estos dos límites son distintos y deben verificarse por separado.Considere la función x al cubo. A medida que x se acerca al infinito positivo, el valor aumenta sin límite.A medida que x se acerca al infinito negativo, el valor disminuye sin límite.Por el contrario, la función seno oscila entre −1 y 1. Dado que nunca se asienta, su límite en el infinito no existe.Algunas funciones se acercan a un valor finito, como uno dividido por x más 2. Como x tiende a infinito, uno sobre x se convierte en cero, dejando el valor 2. Esta línea horizontal, y igual a 2, se llama asíntota horizontal.Este concepto aparece en circuitos reales, como cuando se carga un condensador en un circuito RC en serie.Cuando se conecta una batería, la carga del condensador aumenta con el tiempo. Tomando el límite a medida que el tiempo t se acerca al infinito, el término exponencial será cero y la carga del condensador se acercará a un valor máximo constante, que representa la asíntota horizontal de la curva.

límites en el infinito

La funci&#243;n que decrece a medida que la entrada se vuelve muy grande proporciona un claro ejemplo de c&#243;mo se comportan las funciones matem&#225;ticas en valores extremos. Cuando la entrada aumenta de forma continua, la salida disminuye progresivamente, aproxim&#225;ndose a un valor fijo determinado. Aunque la salida nunca alcanza ese valor, se acerca a &#233;l sin l&#237;mite. Este comportamiento es fundamental para comprender c&#243;mo se comportan las funciones cuando la entrada crece indefinidamente. La representaci&#243;n gr&#225;fica de esta funci&#243;n muestra que su curva se aplana y parece estabilizarse cerca de una recta horizontal espec&#237;fica. Esta recta, a la que el gr&#225;fico se aproxima pero nunca llega a tocar, representa el comportamiento a largo plazo de la funci&#243;n.
Otro ejemplo de este tipo de comportamiento se observa en una funci&#243;n que se estabiliza gradualmente en valores distintos seg&#250;n aumente o disminuya la entrada sin l&#237;mite. La salida se aproxima a un cierto valor superior cuando la entrada se hace extremadamente grande en la direcci&#243;n positiva. Por el contrario, cuando la entrada se hace muy grande en la direcci&#243;n negativa, la salida se acerca a un valor inferior diferente. La funci&#243;n no cruza ni alcanza estos dos valores de referencia, sino que indica c&#243;mo se comporta a medida que la entrada se aproxima a valores extremos en cualquiera de las dos direcciones. En un gr&#225;fico, la curva se flexiona suavemente y se acerca a esos l&#237;mites superior e inferior, mostrando una transici&#243;n continua sin cambios bruscos.
Este tipo de an&#225;lisis ayuda a comprender el comportamiento de las funciones a largo plazo y es esencial en muchas &#225;reas de las matem&#225;ticas, especialmente para describir tendencias, estimar valores y modelizar fen&#243;menos del mundo real en t&#233;rminos abstractos. Estas caracter&#237;sticas son significativas al analizar la estabilidad o los l&#237;mites de los sistemas representados por funciones matem&#225;ticas.

Límites en el infinito

La funci&#243;n que decrece a medida que la entrada se vuelve muy grande proporciona un claro ejemplo de c&#243;mo se comportan las funciones ...

Education

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JoVE Core

Math Fundamentals

Fundamentos de matemáticas

Unit

Unidad

Limits

Límites

KEY TERMS AND CONCEPTS

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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 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...

limit laws ii

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 Bounds to Find a Limit

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the precise definition of a limit

Epsilon-Delta Logic for Limits

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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.

math fundamentals

Horizontal Asymptotes in Function Graphs

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