Asignaturas

Matemáticas

¡Aprende a Graficar Límites en GeoGebra Fácilmente!

Name: Knowunity
Brand: Knowunity

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¡Aprende a Graficar Límites en GeoGebra Fácilmente!

Yermahin Carreño

@ermahinarreo_lheeotb

63 Seguidores

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The document provides a comprehensive guide on solving limit problems and graphing functions using GeoGebra. It covers various types of limits, including lateral limits, indeterminate forms, limits at infinity, and trigonometric limits. The guide also demonstrates how to use GeoGebra to visualize and verify limit solutions.

Key points:

Detailed step-by-step solutions for various limit problems
Integration of GeoGebra for graphing and verification
Explanation of important trigonometric identities
Application of limits in determining function continuity

5/7/2024

3175

Desarrollo de los ejercicios
Graficar en GeoGebra la siguiente función a trozos, y de acuerdo con ella determinar los límites laterales
dado

Solving Indeterminate Limits

This page demonstrates how to solve an indeterminate limit of the form 0/0. The limit to be calculated is:

lim [2x² + 3x + 1] / (x² - 2x - 3) as x → -1

Definition: An indeterminate form is a limit expression that does not have a immediately obvious value and requires further manipulation to evaluate.

The solution process involves:

Evaluating the limit directly (which results in 0/0)
Factoring both numerator and denominator
Simplifying and canceling common factors
Evaluating the simplified limit

Example: After simplification, the limit becomes (2x + 1) / (x - 3) as x → -1, which evaluates to 1/4 or 0.25.

Ver

Limits at Infinity and GeoGebra Verification

This section covers the calculation of a limit at infinity and its verification using GeoGebra. The limit to be evaluated is:

lim (-6x + x² + 1) / (2x⁴ - x) as x → ∞

The solution process includes:

Dividing both numerator and denominator by the highest power of x in the denominator (x⁴)
Simplifying the resulting expression
Evaluating the limit as x approaches infinity

Highlight: The use of GeoGebra to graph the function and visually confirm the limit's existence is a powerful tool for understanding limit behavior.

Example: The final result of the limit is -3, which can be seen on the GeoGebra graph as the function approaches this value as x increases.

Ver

GeoGebra Visualization of the Continuous Piecewise Function

The final page shows the GeoGebra graph of the piecewise function after determining the values of a and b. The graph visually confirms the continuity of the function at the transition points.

Highlight: GeoGebra proves to be an invaluable tool for visualizing mathematical concepts, especially in the study of limits and continuity.

The complete piecewise function is now defined as:

f(x) = { x + 1, x < 1 4x - 2, 1 ≤ x ≤ 2 3x, x > 2

Example: The graph in GeoGebra shows smooth transitions at x = 1 and x = 2, confirming the function's continuity.

This visual representation helps solidify the understanding of how limits determine the continuity of piecewise functions.

Ver

Trigonometric Limits Without L'Hôpital's Rule

This page focuses on evaluating a trigonometric limit without using L'Hôpital's Rule. The limit to be calculated is:

lim (sec² x) / (tan² x sec x) as x → π/2

The solution involves:

Recalling key trigonometric identities (sec x = 1/cos x, tan x = sin x / cos x)
Substituting these identities into the limit expression
Simplifying and manipulating the expression
Evaluating the final form of the limit

Vocabulary: Trigonometric identities are equations involving trigonometric functions that are true for all values of the variables involved.

Example: After simplification and application of the "ear rule", the limit evaluates to 0.

Ver

Application of Limits in Function Continuity

This section presents an application problem involving limits and continuity. A piecewise function is given:

f(x) = { x + 1, x < 1 ax + b, 1 ≤ x ≤ 2 3x, x ≥ 2

The task is to determine the values of a and b that make the function continuous.

Definition: A function is continuous if it has no breaks, gaps, or jumps in its graph.

The solution process involves:

Setting up equations based on the continuity conditions at x = 1 and x = 2
Solving the system of equations to find a and b
Verifying the solution using GeoGebra

Highlight: This problem demonstrates the practical application of limits in ensuring function continuity, a crucial concept in calculus.

Example: The final solution yields a = 4 and b = -2, which can be verified by graphing the function in GeoGebra.

Ver

Graphing Piecewise Functions and Determining Lateral Limits

This section focuses on graphing a piecewise function in GeoGebra and determining its lateral limits. The function is defined as:

f(x) = { 2x + 3, x ≤ 2 x² + 1, x > 2

The lateral limits to be determined are:

lim f(x) as x → -∞
lim f(x) as x → +∞
lim f(x) as x → 2⁻
lim f(x) as x → 2⁺

Example: For lim f(x) as x → -∞, we evaluate 2(-∞) + 3 = -∞

Highlight: The use of GeoGebra allows for a visual representation of the function, making it easier to understand the behavior of limits.

Vocabulary: Lateral limits refer to the value a function approaches from either the left (-) or right (+) side of a point.

Ver

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Knowunity es la app educativa nº 1 en cinco países europeos

Knowunity fue un artículo destacado por Apple y ha ocupado sistemáticamente los primeros puestos en las listas de la tienda de aplicaciones dentro de la categoría de educación en Alemania, Italia, Polonia, Suiza y Reino Unido. Regístrate hoy en Knowunity y ayuda a millones de estudiantes de todo el mundo.

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Knowunity es la app educativa nº 1 en cinco países europeos

4.9+

valoración media de la app

13 M

estudiantes les encanta Knowunity

#1

en las listas de aplicaciones educativas de 12 países

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estudiantes han subido contenidos escolares

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Usuario de iOS

Me encanta esta app [...] ¡¡¡Recomiendo Knowunity a todo el mundo!!! Pasé de un 2 a un 9 con él :D

Javi, usuario de iOS

La app es muy fácil de usar y está muy bien diseñada. Hasta ahora he encontrado todo lo que estaba buscando y he podido aprender mucho de las presentaciones.

Mari, usuario de iOS

Me encanta esta app ❤️, de hecho la uso cada vez que estudio.

Asignaturas

Matemáticas

¡Aprende a Graficar Límites en GeoGebra Fácilmente!

Yermahin Carreño

@ermahinarreo_lheeotb

63 Seguidores

Seguir

Key points:

Detailed step-by-step solutions for various limit problems
Integration of GeoGebra for graphing and verification
Explanation of important trigonometric identities
Application of limits in determining function continuity

5/7/2024

3175

11/2º Bach

Matemáticas

122

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Solving Indeterminate Limits

This page demonstrates how to solve an indeterminate limit of the form 0/0. The limit to be calculated is:

lim [2x² + 3x + 1] / (x² - 2x - 3) as x → -1

Definition: An indeterminate form is a limit expression that does not have a immediately obvious value and requires further manipulation to evaluate.

The solution process involves:

Evaluating the limit directly (which results in 0/0)
Factoring both numerator and denominator
Simplifying and canceling common factors
Evaluating the simplified limit

Example: After simplification, the limit becomes (2x + 1) / (x - 3) as x → -1, which evaluates to 1/4 or 0.25.

Registrarse

Regístrate para obtener acceso ilimitado a miles de materiales de estudio. ¡Es gratis!

Acceso a todos los documentos

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Mejora tus notas

Al registrarte aceptas las Condiciones del servicio y la Política de privacidad.

Limits at Infinity and GeoGebra Verification

This section covers the calculation of a limit at infinity and its verification using GeoGebra. The limit to be evaluated is:

lim (-6x + x² + 1) / (2x⁴ - x) as x → ∞

The solution process includes:

Dividing both numerator and denominator by the highest power of x in the denominator (x⁴)
Simplifying the resulting expression
Evaluating the limit as x approaches infinity

Highlight: The use of GeoGebra to graph the function and visually confirm the limit's existence is a powerful tool for understanding limit behavior.

Example: The final result of the limit is -3, which can be seen on the GeoGebra graph as the function approaches this value as x increases.

Registrarse

Regístrate para obtener acceso ilimitado a miles de materiales de estudio. ¡Es gratis!

Acceso a todos los documentos

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GeoGebra Visualization of the Continuous Piecewise Function

The final page shows the GeoGebra graph of the piecewise function after determining the values of a and b. The graph visually confirms the continuity of the function at the transition points.

Highlight: GeoGebra proves to be an invaluable tool for visualizing mathematical concepts, especially in the study of limits and continuity.

The complete piecewise function is now defined as:

f(x) = { x + 1, x < 1 4x - 2, 1 ≤ x ≤ 2 3x, x > 2

Example: The graph in GeoGebra shows smooth transitions at x = 1 and x = 2, confirming the function's continuity.

This visual representation helps solidify the understanding of how limits determine the continuity of piecewise functions.

Registrarse

Regístrate para obtener acceso ilimitado a miles de materiales de estudio. ¡Es gratis!

Acceso a todos los documentos

Únete a millones de estudiantes

Mejora tus notas

Al registrarte aceptas las Condiciones del servicio y la Política de privacidad.

Trigonometric Limits Without L'Hôpital's Rule

This page focuses on evaluating a trigonometric limit without using L'Hôpital's Rule. The limit to be calculated is:

lim (sec² x) / (tan² x sec x) as x → π/2

The solution involves:

Recalling key trigonometric identities (sec x = 1/cos x, tan x = sin x / cos x)
Substituting these identities into the limit expression
Simplifying and manipulating the expression
Evaluating the final form of the limit

Vocabulary: Trigonometric identities are equations involving trigonometric functions that are true for all values of the variables involved.

Example: After simplification and application of the "ear rule", the limit evaluates to 0.

Registrarse

Regístrate para obtener acceso ilimitado a miles de materiales de estudio. ¡Es gratis!

Acceso a todos los documentos

Únete a millones de estudiantes

Mejora tus notas

Al registrarte aceptas las Condiciones del servicio y la Política de privacidad.

Application of Limits in Function Continuity

This section presents an application problem involving limits and continuity. A piecewise function is given:

f(x) = { x + 1, x < 1 ax + b, 1 ≤ x ≤ 2 3x, x ≥ 2

The task is to determine the values of a and b that make the function continuous.

Definition: A function is continuous if it has no breaks, gaps, or jumps in its graph.

The solution process involves:

Setting up equations based on the continuity conditions at x = 1 and x = 2
Solving the system of equations to find a and b
Verifying the solution using GeoGebra

Highlight: This problem demonstrates the practical application of limits in ensuring function continuity, a crucial concept in calculus.

Example: The final solution yields a = 4 and b = -2, which can be verified by graphing the function in GeoGebra.

Registrarse

Regístrate para obtener acceso ilimitado a miles de materiales de estudio. ¡Es gratis!

Acceso a todos los documentos

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Al registrarte aceptas las Condiciones del servicio y la Política de privacidad.

Graphing Piecewise Functions and Determining Lateral Limits

This section focuses on graphing a piecewise function in GeoGebra and determining its lateral limits. The function is defined as:

f(x) = { 2x + 3, x ≤ 2 x² + 1, x > 2

The lateral limits to be determined are:

lim f(x) as x → -∞
lim f(x) as x → +∞
lim f(x) as x → 2⁻
lim f(x) as x → 2⁺

Example: For lim f(x) as x → -∞, we evaluate 2(-∞) + 3 = -∞

Highlight: The use of GeoGebra allows for a visual representation of the function, making it easier to understand the behavior of limits.

Vocabulary: Lateral limits refer to the value a function approaches from either the left (-) or right (+) side of a point.

Registrarse

Regístrate para obtener acceso ilimitado a miles de materiales de estudio. ¡Es gratis!

Acceso a todos los documentos

Únete a millones de estudiantes

Mejora tus notas

Al registrarte aceptas las Condiciones del servicio y la Política de privacidad.

Registrarse

Regístrate para obtener acceso ilimitado a miles de materiales de estudio. ¡Es gratis!

Acceso a todos los documentos

Únete a millones de estudiantes

Mejora tus notas

Al registrarte aceptas las Condiciones del servicio y la Política de privacidad.

¿No encuentras lo que buscas? Explora otros temas.

Knowunity es la app educativa nº 1 en cinco países europeos

Knowunity fue un artículo destacado por Apple y ha ocupado sistemáticamente los primeros puestos en las listas de la tienda de aplicaciones dentro de la categoría de educación en Alemania, Italia, Polonia, Suiza y Reino Unido. Regístrate hoy en Knowunity y ayuda a millones de estudiantes de todo el mundo.

Descargar en

Google Play

Descargar en

App Store

4.9+

valoración media de la app

13 M

estudiantes les encanta Knowunity

#1

en las listas de aplicaciones educativas de 12 países

950 K+

estudiantes han subido contenidos escolares

¿Aún no estás convencido? Mira lo que dicen tus compañeros...

Usuario de iOS

Me encanta esta app [...] ¡¡¡Recomiendo Knowunity a todo el mundo!!! Pasé de un 2 a un 9 con él :D

Javi, usuario de iOS

La app es muy fácil de usar y está muy bien diseñada. Hasta ahora he encontrado todo lo que estaba buscando y he podido aprender mucho de las presentaciones.

Mari, usuario de iOS