Name: limits in graphs and real-world change
Uploaded: 2025-11-26T05:43:45.000Z
Duration: 1 min 30 s
Channel: JoVE

Los límites describen el valor al que se acerca un sistema a medida que cambian las entradas, incluso cuando el valor exacto no está definido o es difícil de calcular. Una variación de los valores de salida e entrada muestra que a medida que x se acerca a 1, la salida se acerca a 0,5.El movimiento de vehículos en una carretera ilustra bien este concepto. El tráfico ligero permite un viaje suave y de alta velocidad.A medida que más vehículos ingresan a la carretera, aumenta la congestión y la velocidad promedio comienza a disminuir.A altas densidades, la velocidad continúa disminuyendo y se acerca mucho a cero.Esto modela un límite: la velocidad se acerca a cero, lo que representa un atasco, incluso si nunca llega a cero.Esta relación se describe mediante la fórmula: la velocidad es igual a 100 veces el logaritmo natural de 200 dividido por la densidad.Los datos numéricos muestran que, inicialmente, el aumento de la densidad de vehículos provoca reducciones notables de la velocidad, aunque cada vehículo agregado tiene un pequeño efecto.Más allá de cierto punto, cada vehículo agregado provoca una mayor caída en la velocidad, acercándose rápidamente al embotellamiento.Un gráfico de velocidad versus densidad de vehículos muestra una curva inclinada hacia abajo que se aplana cerca de cero.Esta curva representa visualmente un límite: la velocidad se acerca a cero, pero nunca lo alcanza del todo.

determinación numérica y gráfica de límites

Un l&#237;mite describe el valor al que se aproxima una funci&#243;n a medida que su entrada se acerca a un punto determinado. Incluso cuando una funci&#243;n no est&#225; definida en un valor espec&#237;fico, los l&#237;mites permiten analizar su comportamiento cerca de ese punto. Este concepto es fundamental en el c&#225;lculo y esencial para comprender la continuidad, las derivadas y las integrales.
Matem&#225;ticamente, una funci&#243;n f(x) tiene un l&#237;mite L en x = a si f(x) tiende a L cuando x se aproxima arbitrariamente a a. Esto se escribe como:
40100_Equation_1.svg
Esta notaci&#243;n expresa que la funci&#243;n tiende a L independientemente de si f(a) est&#225; definida.
El concepto de l&#237;mite se puede ilustrar con el ejemplo del flujo del tr&#225;fico. Cuando las carreteras est&#225;n despejadas, los veh&#237;culos circulan libremente a mayor velocidad. A medida que aumenta la densidad del tr&#225;fico, la velocidad disminuye. En situaciones de congesti&#243;n extrema, la velocidad tiende a cero, aunque rara vez se detiene por completo. El l&#237;mite de la velocidad al aumentar la densidad demuestra c&#243;mo se comportan las funciones bajo restricciones sin necesidad de c&#225;lculos exactos en cada punto.
Los gr&#225;ficos proporcionan una forma intuitiva de comprender los l&#237;mites. Al representar una funci&#243;n cerca de un punto de inter&#233;s, podemos observar la tendencia de sus valores. Incluso si una funci&#243;n presenta una discontinuidad, el l&#237;mite puede seguir existiendo. Esto es &#250;til para comprender las tendencias en sistemas f&#237;sicos, econ&#243;micos y de ingenier&#237;a.
Los l&#237;mites son fundamentales para definir las derivadas, que miden c&#243;mo cambian las funciones. Tambi&#233;n forman la base de las integrales, que cuantifican la acumulaci&#243;n. Mediante la aplicaci&#243;n de t&#233;cnicas num&#233;ricas, gr&#225;ficas y anal&#237;ticas, los l&#237;mites ayudan a describir sistemas din&#225;micos, optimizar procesos y resolver problemas matem&#225;ticos complejos eficazmente.

Determinación numérica y gráfica de límites

Un l&#237;mite describe el valor al que se aproxima una funci&#243;n a medida que su entrada se acerca a un punto determinado. Incluso cuando una ...

Education

Educación

JoVE Core

Math Fundamentals

Fundamentos de matemáticas

Unit

Unidad

Limits

Límites

KEY TERMS AND CONCEPTS

TÉRMINOS Y CONCEPTOS CLAVE

Introduction to Limits

limits in graphs and real-world change

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 trend of the function near a point, not just the value at one exact input.

A traffic example helps show the idea clearly. When roads are clear, vehicles move faster. As traffic density increases, speed drops. In heavy congestion, speed may approach zero even if traffic does not fully stop.

Graphs also make limits easier to see. By looking at a graph near a point of interest, students can track how the function behaves. Even if the graph has a discontinuity, the limit may still exist. This is useful for studying patterns in physical, economic, and engineering systems.

Limits also support later calculus topics. They help define derivatives, which measure how a function changes. They also form the basis of integrals, which measure accumulation. Using numerical, graphical, and analytical methods, limits help describe changing systems and solve complex math problems.

Limits in Graphs and Real-World Change

tangent to a curve

Secant Lines Approaching a Tangent

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...

velocity of an object

Finding Instantaneous Velocity from Motion

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

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

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...

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...

evaluating limits by direct substitution

Using Continuity to Find Limits

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...

the squeeze theorem

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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continuity of a function

Types of Function Discontinuities

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properties of continuous functions

How Continuous Functions Stay Smooth

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the intermediate value theorem

Using Continuity to Find Roots

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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...

key terms and concepts

Learn how limits describe function behavior near a point, use graph and traffic examples, and see why limits matter for derivatives and integrals.

Limits in Graphs and Real-World Change | 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

Limits in Graphs and Real-World Change

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