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08/09/12 MAD: Unidad 2
1/1148.247.220.103/GAP2012/mod/scorm/player.php?a=174¤torg=ORG-3C12D73B277754E97187…
En la unidad anterior viste el concepto defunción y algunos de los diferentes tipos, asícomo su aplicación para describir elcomportamiento de las diversas situacionesque se presentan dentro del área económico-administrativa, tales como ingresos, costos,utilidades y punto de equilibrio.
En el presente tema estudiarás el conceptode límite y cómo describe de forma precisael comportamiento de una función cuando losvalores de la variable independiente estánmuy próximos, aunque nunca igual, a un valorconstante y su utilidad en los procesoseconómico-administrativos tales comorendimiento y producción máxima; asímismo, determinarás cuando una función escontinua y su aplicación en procesosproductivos y su impacto en los costos deproducción.
Unidad 2. Límites y continuidad
Presentación de la unidad
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08/09/12 MAD: Unidad 2
1/1148.247.220.103/GAP2012/mod/scorm/player.php?a=174¤torg=ORG-3C12D73B277754E97187…
En esta unidad:
Identificarás los elementos del álgebrade límites y teórico práctico de lacontinuidad de una función.Aplicarás los elementos del álgebra delímites para determinar el alcance de unproceso desde el punto de vistaeconómico-administrativo.Calcularás la continuidad de unafunción en relación a los puntos en losque el proceso de producción presentauna tendencia diferente de costos.
Unidad 2. Límites y continuidad
Propósitos de la unidad
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08/09/12 MAD: Unidad 2
1/1148.247.220.103/GAP2012/mod/scorm/player.php?a=174¤torg=ORG-3C12D73B277754E97187…
Determina el comportamiento de unafunción de acuerdo a los límites de lamisma para conocer su impacto en losprocesos económico-administrativos,mediante la aplicación de los teoremas delímites de una función y su relación confunciones continuas y discontinuas.
Unidad 2. Límites y continuidad
Competencia específica
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08/09/12 MAD: Unidad 2
1/1148.247.220.103/GAP2012/mod/scorm/player.php?a=174¤torg=ORG-3C12D73B277754E97187…
Con el álgebra de límites podemosaproximarnos a un valor ya sea un númerocualquiera o el infinito de manera exacta.
Así podríamos decir que el límite de unafunción describe el comportamiento de unafunción f(x) conforme la variableindependiente se aproxima a un valorconstante.
Unidad 2. Límites y continuidad
2.1. Álgebra de límites
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1
Fichero Álgebra de límites
Ejemplo Se tiene la función
y se requiere determinar su comportamiento cuando los valores de x tienden o se acercan a 1. Solución Podemos observar que la función no está definida en x = 1, esto es que cuando x toma el valor de 1, la función tiende al infinito:
Ya que cualquier número dividido entre cero es igual a infinito. Sin embargo si podemos determinar el comportamiento o valores que va tomando la función cuando x → 1 (x tiende a 1) ya sea con valores más pequeños a uno o bien más grandes a uno: 1 > x > 1.
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Gráfica
Conclusión Como pudiste observar, conforme x se acerca a 1 la función es igual a ± 20000, dependiendo de si se va acercando por la derecha o por la izquierda a 1, es decir:
En conclusión, tenemos que: Cuando f(x) se acerca cada vez más a un número Límite (C), conforme x se aproxima a un valor constante “a” por cualquier lado, entonces C será el límite de la función y se escribe:
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1
Operación para determinar los límites de una función Solución: Desarrollo
En la presente fórmula de álgebra de límites podemos observar que al evaluar el límite de una función sea f(x), g(x), o cualquiera, se tiene que sustituir en dicha función el valor del número hacia el cual tiende x y el resultado será siempre un número constante.
En esta segunda fórmula observamos que al sumar o restar
dos funciones evaluadas en el valor hacia el cual tiende x,
podemos obtener de manera independiente el resultado de cada
función al sustituir el valor de x en cada una y finalmente sumar
o restar ambos resultados según sea el caso.
Para este caso, tenemos que al multiplicar dos funciones evaluadas en el valor hacia el cual tiende x, podemos obtener de manera independiente el resultado de cada función al sustituir el valor de x en cada una y finalmente multiplicar ambos resultados.
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En la siguiente fórmula, tenemos que al dividir dos funciones
evaluadas en el valor hacia el cual tiende x, podemos obtener de
manera independiente el resultado de cada función al sustituir el
valor de x en cada una y finalmente dividir ambos resultados.
En esta fórmula, el valor hacia el cual tiende x se sustituye en la función y posteriormente se evaluará la potencia a la que está elevada la función.
En ésta última fórmula de igual manera sustituyes el valor de x
en la función y posteriormente calculas la raíz n-sima a la que
está elevada x.
CONCLUSIÓN:
El evaluar un límite de una función es tan simple como
sustituir el valor hacia el cual tiende x en la función.
El aplicar el álgebra de límites es únicamente aplicar las
operaciones de suma, resta, multiplicación, división, raíz n-
sima y elevar a una potencia en los resultados obtenidos de
las funciones a las que ya se les ha sustituido el valor límite.
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Límite de una función constante Si se tiene una función constante f(x) = C, el límite de la función
cuando x tienda a un valor “a”, será siempre C, esto es:
Ejemplo: determina el límite de la función: f(x) = 10, cuando x → 8. Solución: siguiendo el razonamiento de la fórmula general anterior, se tiene que:
Límite de una función idéntica
Si se considera la función idéntica f(x) = x, cuando x tiende a un valor “a”, su límite será siempre el valor constante “a”, es decir:
Ejemplo: determina el límite de la función: f(x) = x, cuando x → 3.
Solución: siguiendo el razonamiento de la fórmula general anterior, se tiene que:
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Límites infinitos
Cuando se tiene una función racional
en la que q(x) se hace cero cuando x tiende a un valor constante “a”, entonces, f(x) = ∞, es decir:
Ejemplo: determina el límite de la función: cuando x → 1.
Solución: sustituyendo el valor de x, en la función se tiene que:
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3
Límite de cualquier función
Para cualquier función f(x), se tiene que el límite de la función cuando x → a, el límite es el número constante que resulta de sustituir el valor de “a” en la función. Ejemplo 1:
Determina el límite de la función: f(x) = , cuando x → 0 y
cuando x → 5.
Solución: sustituyendo el valor de x, en la función para cada caso se
tiene que:
Ejemplo 2. : determina el límite de la función: cuando x → 0
y cuando x → 2.
Solución: sustituyendo el valor de x, en la función para cada caso se tiene que:
Y
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08/09/12 MAD: Unidad 2
1/1148.247.220.103/GAP2012/mod/scorm/player.php?a=174¤torg=ORG-3C12D73B277754E97187…
Cuando x → ∞, el valor de la función puede crecer o decrecer indefinidamente, sinembargo existen casos en los que la función adquiere valores reales. A continuaciónveremos un ejemplo para el cálculo de límite de una función cuando la variableindependiente tiende al infinito.
Ejemplo. ¿Cuál será el límite de ?
Solución: al aplicar el limite infinito en la función, se tiene que:
Unidad 2. Límites y continuidad
2.1.2. Límites de una función cuando la variable tiende al infinito
Da clic en la flecha para continuar. 1 de 2
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08/09/12 MAD: Unidad 2
1/1148.247.220.103/GAP2012/mod/scorm/player.php?a=174¤torg=ORG-3C12D73B277754E97187…
En el siguiente documento te presentamos dos ejemplos de los límites de una funcióncuando la variable tiende al infinito, descarga el documento para que lo revises.
Da clic en el icono para descargar el documento.Límite de una función cuando lavariable tiende al infinito.
Unidad 2. Límites y continuidad
2.1.2. Límites de una función cuando la variable tiende al infinito
Da clic en la Actividad 1: Maximización de costo promedio. 2 de 2
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Matemáticas Administrativas Unidad 2. Límites y continuidad
Educación Superior Abierta y a Distancia • Ciencias Sociales y Administrativas 1
CUADERNILLO DE EJERCICIOS LOS LÍMITES Y APLICACIÓN EN FUNCIONES
CUADERNILLO DE EJERCICIOS: Los límites y aplicación en funciones CARRERA: Licenciatura en Matemáticas CUATRIMESTRE: Dos
ASIGNATURA: Matemáticas Administrativas ELABORÓ/REVISÓ: Nalleli Guadalupe María Acosta Topete / Alicia Pérez Godínez
UNIDAD: Límites y continuidad
Fórmulas básicas
Fórmula / Símbolo Descripción Fórmula / Símbolo Descripción
Ley de signos para multiplicación
Menor que Mayor que
Menor o igual que Mayor o igual que
Aproximadamente igual Aproximadamente Diferente que (a)
Igual que (a) Infinito
Incremento, gradiente, cambio Que tiende a… /que se aproxima
a… Porciento
Raíz cuadrada Raíz cúbica
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Matemáticas Administrativas Unidad 2. Límites y continuidad
Educación Superior Abierta y a Distancia • Ciencias Sociales y Administrativas 2
Fórmulas unidad 2. Limites y Continuidad
Fórmula / Símbolo Descripción Fórmula / Símbolo Descripción
1.
2.
3.
4.
5.
Álgebra de límites para dos funciones cuya variable independiente tiene a un valor a:
y
Límite de una función constante
Límite de una función idéntica
Cuando
Entonces
Límites infinitos
1. Una función será continua si f(x) está definida en x = a, es decir, que sus valores son reales.
2. Una función será continua si el Límite de la función f(x) cuando x → a existe.
3. Una función será continua si:
Condiciones para comprobar la continuidad de una función
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Matemáticas Administrativas Unidad 2. Límites y continuidad
Educación Superior Abierta y a Distancia • Ciencias Sociales y Administrativas 3
Ejemplo:
Los ingresos obtenidos por un comercial en una radiodifusora están dados por la siguiente función:
En donde representa las semanas que es transmitido el comercial. Determine cuales serán los ingresos que se pueden lograr
si el comercial se mantiene al aire por tiempo indefinido.
Solución: Ya que se desea saber cuáles serán los ingresos si se mantiene al aire por tiempo indefinido el comercial, entonces es necesario
calcular la tendencia de la función de ingresos cuando el tiempo tiende a .
Para evaluar el límite de una función racional en el infinito primero es necesario dividir a toda la función entre la mayor potencia, en este caso :
Evaluando ahora el límite cuando :
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Matemáticas Administrativas Unidad 2. Límites y continuidad
Educación Superior Abierta y a Distancia • Ciencias Sociales y Administrativas 4
Ahora bien, cada valor dividido entre infinito será cero, ya que implica dividir cualquier valor entre un número muy grande,
(ejemplo: divide una rebanada de pastel entre toda la población del D.F., a cada persona no le toca ni siquiera una migaja), lo
que es muy similar a dividir entre el infinito:
Y como la función está dada en miles de dólares:
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Matemáticas Administrativas Unidad 2. Límites y continuidad
Educación Superior Abierta y a Distancia • Ciencias Sociales y Administrativas 5
Ejercicio 1. Maximización de costo promedio
El costo promedio mensual debido al departamento de recursos humanos en un buffet de abogados está dado por la siguiente función:
En donde representa al personal del departamento de dicho departamento y se desea aumentar el número de empleados en el área de
recursos humanos. Determine el costo promedio máximo que puede tener el buffet si aumenta el número de empleados.
Respuesta: $__________________
Solución:
Conclusión:
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08/09/12 MAD: Unidad 2
1/1148.247.220.103/GAP2012/mod/scorm/player.php?a=174¤torg=ORG-3C12D73B277754E97187…
Para que practiques y refuerces tus conocimientos de este tema, realiza el siguienteejercicio.
1. Descarga el archivo Cuadernillo de ejercicios: Los límites y aplicación enfunciones.
2. Resuelve el ejercicio 1 Maximización de costo promedio. En el documentoencontrarás un formulario y un ejemplo que te podrán orientar para elaborar elEjercicio. Lee con mucha atención el planteamiento que se te proporciona.
3. Guarda tu documento como MA_U2_CP01_XXYZ, en formato Word 97-2003, yenvíalo a tu Facilitador(a) a través de la sección de Tareas. Sustituye las XXpor las dos primeras letras de tu primer nombre, la Y por la inicial de tu apellidopaterno y la Z por la inicial de tu apellido materno.
Da clic en el icono para descargar el documento "Los límites y aplicación enfunciones"
Unidad 2. Límites y continuidad
Actividad 1: Maximización de costo promedio
Da clic en la flecha para continuar. 1 de 2
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Matemáticas Administrativas Unidad 2. Límites y continuidad Actividad 1. Maximización del costo promedio
Educación Superior Abierta y a Distancia • Ciencias Sociales y Administrativas 1
CUADERNILLO DE EJERCICIOS LOS LÍMITES Y APLICACIÓN EN FUNCIONES
CUADERNILLO DE EJERCICIOS: Los límites y aplicación en funciones CARRERA: Licenciatura en Matemáticas CUATRIMESTRE: Dos
ASIGNATURA: Matemáticas Administrativas ELABORÓ/REVISÓ: Nalleli Guadalupe María Acosta Topete / Alicia Pérez Godínez
UNIDAD: Límites y continuidad
Fórmulas básicas Fórmula / Símbolo Descripción Fórmula / Símbolo Descripción
Ley de signos para multiplicación
Menor que Mayor que
Menor o igual que Mayor o igual que
Aproximadamente igual Aproximadamente Diferente que (a)
Igual que (a) Infinito
Incremento, gradiente, cambio Que tiende a… /que se aproxima
a… Porciento
Raíz cuadrada Raíz cúbica
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Matemáticas Administrativas Unidad 2. Límites y continuidad Actividad 1. Maximización del costo promedio
Educación Superior Abierta y a Distancia • Ciencias Sociales y Administrativas 2
Fórmulas unidad 2. Limites y Continuidad
Fórmula / Símbolo Descripción Fórmula / Símbolo Descripción 1.
2.
3.
4.
5.
Álgebra de límites para dos funciones cuya variable independiente tiene a un valor a:
y
Límite de una función constante
Límite de una función idéntica
Cuando
Entonces
Límites infinitos
1. Una función será continua si f(x) está definida en x = a, es decir, que sus valores son reales.
2. Una función será continua si el Límite de la función f(x) cuando x → a existe.
3. Una función será continua si:
Condiciones para comprobar la continuidad de una función
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Matemáticas Administrativas Unidad 2. Límites y continuidad Actividad 1. Maximización del costo promedio
Educación Superior Abierta y a Distancia • Ciencias Sociales y Administrativas 3
Ejemplo:
Los ingresos obtenidos por un comercial en una radiodifusora están dados por la siguiente función:
En donde representa las semanas que es transmitido el comercial. Determine cuales serán los ingresos que se pueden lograr
si el comercial se mantiene al aire por tiempo indefinido.
Solución: Ya que se desea saber cuáles serán los ingresos si se mantiene al aire por tiempo indefinido el comercial, entonces es necesario
calcular la tendencia de la función de ingresos cuando el tiempo tiende a .
Para evaluar el límite de una función racional en el infinito primero es necesario dividir a toda la función entre la mayor potencia, en este caso :
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Matemáticas Administrativas Unidad 2. Límites y continuidad Actividad 1. Maximización del costo promedio
Educación Superior Abierta y a Distancia • Ciencias Sociales y Administrativas 4
Evaluando ahora el límite cuando :
Ahora bien, cada valor dividido entre infinito será cero, ya que implica dividir cualquier valor entre un número muy grande,
(ejemplo: divide una rebanada de pastel entre toda la población del D.F., a cada persona no le toca ni siquiera una migaja), lo
que es muy similar a dividir entre el infinito:
Y como la función está dada en miles de dólares:
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Matemáticas Administrativas Unidad 2. Límites y continuidad Actividad 1. Maximización del costo promedio
Educación Superior Abierta y a Distancia • Ciencias Sociales y Administrativas 5
Ejercicio 1. Maximización de costo promedio
El costo promedio mensual debido al departamento de recursos humanos en un buffet de abogados está dado por la siguiente función:
En donde representa al personal del departamento de dicho departamento y se desea aumentar el número de empleados en el área de
recursos humanos. Determine el costo promedio máximo que puede tener el buffet si aumenta el número de empleados.
Respuesta: $__________________
Solución:
Conclusión:
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08/09/12 MAD: Unidad 2
1/1148.247.220.103/GAP2012/mod/scorm/player.php?a=174¤torg=ORG-3C12D73B277754E97187…
Para ingresar a la sección de Tareas: En la ruta (parte superior izquierda del aula) daclic en Matemáticas. Se enlistarán las actividades de la unidad, da clic en la Actividad 1:Maximización de costo promedio.
Unidad 2. Límites y continuidad
Actividad 1: Maximización de costo promedio
Al terminar la actividad, da clic en la Actividad 2. Foro. Maximización costo promedio. 2 de 2
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08/09/12 MAD: Unidad 2
1/1148.247.220.103/GAP2012/mod/scorm/player.php?a=174¤torg=ORG-3C12D73B277754E97187…
1. Ingresa al Foro: Maximización de costo promedio y responde a la siguiente pregunta:
¿Qué elementos consideraste para resolver el problema de “Maximización decosto promedio”?
2. Revisa las participaciones de tus compañeros(as) y opina acerca de ellas.
3. Consulta la rúbrica de Foro. Recuerda que todas las actividades son consideradas entu promedio final.
Da clic en el icono para descargar el documento.
Unidad 2. Límites y continuidad
Actividad 2. Foro: Maximización de costo promedio
Al concluir tu participación en el foro, da clic el Tema 2.2 Funciones continuas ydiscontinuas.
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08/09/12 MAD: Unidad 2
1/1148.247.220.103/GAP2012/mod/scorm/player.php?a=174¤torg=ORG-3C12D73B277754E97187…
¿A qué nos referimos con las funciones continuas y discontinuas?
Unidad 2. Límites y continuidad
2.2. Funciones continuas y discontinuas
Da clic en el subtema 2.2.1. Conceptos relacionados a la continuidad y discontinuidad enuna función.
1 de 1
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08/09/12 MAD: Unidad 2
1/1148.247.220.103/GAP2012/mod/scorm/player.php?a=174¤torg=ORG-3C12D73B277754E97187…
Al realizar el cálculo de límites de una función, no siempre el límite coincide con el valorde la función en el punto hacia el cual se aproxima la variable independiente, esto es fácilde detectar al graficar la función en los valores cercanos al límite, ya que la gráfica de lafunción se puede cortar o tener una interrupción en algún punto cercano al límite.
Observa el ejemplo de la siguiente pantalla.
Unidad 2. Límites y continuidad
2.2.1. Conceptos relacionados a la continuidad y discontinuidad en una función
Da clic en la flecha para continuar. 1 de 4
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Gráfica continua y discontinua Observa con atención las siguientes gráficas: Gráfica inicial:
Analiza el comportamiento de la línea morada.
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Analiza el comportamiento de la línea verde
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Observa el punto exactamente cuándo x = 1 en que la gráfica de la línea ya no continúa con el resto de los valores, es decir que hay una ruptura en la gráfica.
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4
Podemos definir que una función es continua cuando no se presenta un corte en la línea que representa su gráfica, mientras que una función es discontinua cuando se presentan cortes en la línea que representa la gráfica de la función.
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08/09/12 MAD: Unidad 2
1/1148.247.220.103/GAP2012/mod/scorm/player.php?a=174¤torg=ORG-3C12D73B277754E97187…
Como pudiste observar en la animación, la continuidad y discontinuidad en una funcióndependerá del corte que se le haga a la línea o no.
Así se tienen tres condiciones que nos permiten descubrir si una función es continua odiscontinua:
Unidad 2. Límites y continuidad
2.2.1. Conceptos relacionados a la continuidad y discontinuidad en una función
Da clic en las fechas para avanzar o retroceder. 3 de 4
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08/09/12 MAD: Unidad 2
1/1148.247.220.103/GAP2012/mod/scorm/player.php?a=174¤torg=ORG-3C12D73B277754E97187…
Recuerda:
Unidad 2. Límites y continuidad
2.2.1. Conceptos relacionados a la continuidad y discontinuidad en una función
Da clic en el subtema 2.2.2. Aplicaciones de funciones continuas y discontinuas. 4 de 4
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08/09/12 MAD: Unidad 2
1/1148.247.220.103/GAP2012/mod/scorm/player.php?a=174¤torg=ORG-3C12D73B277754E97187…
Ahora que sabes cómo es el procedimiento de las funciones continuas y discontinuas,es necesario que conozcas cómo se aplican. Para esto descarga el archivo PDFllamado Ejemplo oferta y demanda y lee con mucha atención el procedimiento que sedescribe. Te sugerimos que en tu libreta de notas vayas elaborando el mismoprocedimiento y compares resultados.
Da clic en el ícono para descargar el documento “Ejemplo oferta y demanda”.
Unidad 2. Límites y continuidad
2.2.2. Aplicaciones de funciones continuas y discontinuas
Da clic en la Actividad 3: Costo total. 1 de 1
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1Educación Superior Abierta y a Distancia • Ciencias Sociales y Administrativas
Matemáticas AdministrativasUnidad 2. Límites y continuidad2.2.2 Aplicaciones de funciones continuas y discontinuas
Ejemplo de oferta y demanda.
Un vendedor de aceites orgánicos en frascos de 250 ml, vende aceite de uva a $90.00 cada frasco, pero si le compran más de 5 frascos, el precio por frasco es de $68.00. ¿Qué se le podría recomendar al vendedor para que pueda conservar su escala de precios de mayoreo sin que se le presenten problemas económicos con su promoción?
Solución:
Si definimos como p(x) a la función de precio de x frascos de aceite de uva, se tiene la siguiente función:
El modelo gráfico que representa a esta función de oferta vs demanda es:
Con lo que se observa que el precio de 10 frascos es de $900.00, y de 11 frascos es de $935.00, por lo que para evitar contradicciones, el precio de 11 frascos debe ser superior al de 10; si decimos que p es el precio de cada frasco de aceite cuando se compran más de 10 frascos, se debe cumplir que 11p > 900, es decir, p > 900/11 = $81.81 pesos, por lo tanto, el vendedor debe asignar un precio superior a $81.81 pesos para cada frasco cuando le compren más de 10 frascos de aceite de uva.
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08/09/12 MAD: Unidad 2
1/1148.247.220.103/GAP2012/mod/scorm/player.php?a=174¤torg=ORG-3C12D73B277754E97187…
Realiza el siguiente ejercicio.
1. Descarga el documento Cuadernillo de ejercicios: Los límites y aplicación enfunciones.
2. Resuelve el ejercicio 2 Costo total que se te presenta. En ese documentoencontrarás un formulario y un ejemplo que te orientarán. Lee con mucha atención elplanteamiento que se te proporciona.
3. Guarda tu documento como MA_U2_CT02_XXYZ, en formato Word 97-2003, yenvíalo a tu Facilitador(a) a través de la sección de Tareas. Sustituye las XX por lasdos primeras letras de tu primer nombre, la Y por la inicial de tu apellido paterno y la Zpor la inicial de tu apellido materno.
Da clic en el ícono para descargar el documento “Los límites y aplicaciónen funciones”.
Para ingresar a la sección de Tareas: En la ruta (parte superior izquierda del aula) da clicen Matemáticas Administrativas. Se enlistarán las actividades de la unidad, da clic en laActividad 3: Costo total.
Unidad 2. Límites y continuidad
Actividad 3: Costo total
Da clic en la Autoevaluación. 1 de 1
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Matemáticas Administrativas Unidad 2.1. Límites y continuidad 2.1.2 Límites de una función cuando la variable tiende al infinito
Educación Superior Abierta y a Distancia • Ciencias Sociales y Administrativas
1
CUADERNILLO DE EJERCICIOS: Los límites y aplicación en funciones
CARRERA: Licenciatura en Matemáticas CUATRIMESTRE: Dos
ASIGNATURA: Matemáticas Administrativas ELABORÓ/REVISÓ: Nalleli Guadalupe María Acosta Topete / Alicia Pérez Godínez
UNIDAD: Límites y Continuidad
Fórmulas básicas Fórmula / Símbolo Descripción Fórmula / Símbolo Descripción
Ley de signos para multiplicación
Menor que Mayor que
Menor o igual que Mayor o igual que
Aproximadamente igual Aproximadamente Diferente que (a)
Igual que (a) Infinito
Incremento, gradiente, cambio Que tiende a… /que se aproxima
a… Porciento
Raíz cuadrada Raíz cúbica
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Matemáticas Administrativas Unidad 2.1. Límites y continuidad 2.1.2 Límites de una función cuando la variable tiende al infinito
Educación Superior Abierta y a Distancia • Ciencias Sociales y Administrativas
2
Fórmulas unidad 2. Limites y Continuidad
Fórmula / Símbolo Descripción Fórmula / Símbolo Descripción 1.
2.
3.
4.
5.
Álgebra de límites para dos funciones cuya variable independiente tiene a un valor a:
y
Límite de una función constante
Límite de una función idéntica
Cuando
Entonces
Límites infinitos
1. Una función será continua si f(x) está definida en x = a, es decir, que sus valores son reales.
2. Una función será continua si el Límite de la función f(x) cuando x → a existe.
3. Una función será continua si:
Condicione para comprobar la continuidad de una función
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Matemáticas Administrativas Unidad 2.1. Límites y continuidad 2.1.2 Límites de una función cuando la variable tiende al infinito
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Ejemplo:
Los ingresos obtenidos por un comercial en una radiodifusora están dados por la siguiente función:
En donde representa las semanas que es transmitido el comercial. Determina cuáles serán los ingresos que se pueden lograr
si el comercial se mantiene al aire por tiempo indefinido.
Solución: Ya que se desea saber cuáles serán los ingresos si se mantiene al aire por tiempo indefinido el comercial, entonces es necesario
calcular la tendencia de la función de ingresos cuando el tiempo tiende a .
Para evaluar el límite de una función racional en el infinito primero es necesario dividir a toda la función entre la mayor potencia, en este caso :
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Matemáticas Administrativas Unidad 2.1. Límites y continuidad 2.1.2 Límites de una función cuando la variable tiende al infinito
Educación Superior Abierta y a Distancia • Ciencias Sociales y Administrativas
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Evaluando ahora el límite cuando :
Ahora bien cada valor dividido entre infinito será cero, ya que implica dividir cualquier valor entre un número muy grande,
(ejemplo: divide una rebanada de pastel entre toda la población del D.F., a cada persona no le toca ni siquiera una migaja), lo
que es muy similar a dividir entre el infinito:
Y como la función está dada en miles de dólares:
Actividad 2. Costo Total
Una empresa dedicada a la fabricación de productos químicos tuvo un derrame de un contaminante químico en un río de la localidad, se
ha determinado por el departamento de manejo de residuos en conjunto con el departamento de administración y finanzas que el costo
generado por dicho derrame estará en función del tiempo que permanezca activo el químico en el agua, lo que está dado por la siguiente
función en miles de pesos:
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Determine cuál será el costo para la empresa conforme pasa el tiempo.
Respuesta: $______________________
Solución:
Conclusión
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08/09/12 MAD: Unidad 2
1/1148.247.220.103/GAP2012/mod/scorm/player.php?a=174¤torg=ORG-3C12D73B277754E97187…
Has llegado al final de la unidad, ingresa a Autoevaluación, lee las preguntas yselecciona la opción correcta.
Para ingresar a la Autoevaluación: En la ruta (parte superior izquierda del aula) da clicen Matemáticas. Se enlistarán las actividades de la unidad, da clic en Autoevaluación.
Unidad 2. Límites y continuidad
Autoevaluación
Al terminar la actividad, da clic en Evidencias de aprendizaje: Álgebra de límites ycontinuidad.
1 de 1
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Para evaluar la unidad, realiza lo siguiente:
1. Descarga el documento en Word llamado Álgebra de límites y continuidad, que seencuentra en la siguiente pantalla.
2. Ese documento está conformado por dos ejercicios:
Ejercicio 1. Cálculo de límites el cual consiste en relacionar columnas. Parasaber la respuesta correcta deberás elaborar los cálculos que se necesitan.Ejercicio 2. Rentabilidad con límites al infinito en el que deberás analizar yelaborar el problema que se te presenta en el documento. Apóyate del editorde fórmulas que tienes en tu computadora.
3. Guarda tu trabajo como MA_U2_EV_XXYZ, en formato Word 97-2003, y colócalo en elPortafolio de evidencias para que el (la) Facilitador(a) lo revise y te retroalimente.Sustituye las XX por las dos primeras letras de tu primer nombre, la Y por la inicial detu apellido paterno y la Z por la inicial de tu apellido materno.
Unidad 2. Límites y continuidad
Evidencias de aprendizaje: Álgebra de límites y continuidad
Da clic en la flecha para continuar. 1 de 2
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08/09/12 MAD: Unidad 2
1/1148.247.220.103/GAP2012/mod/scorm/player.php?a=174¤torg=ORG-3C12D73B277754E97187…
4. Consulta la escala de evaluación para esta evidencia de aprendizaje.
Da clic en el icono para descargar el documento “Algebra de límites ycontinuidad”.
Da clic en el icono para descargar el documento “Escala de evaluación”.
Para ingresar al Portafolio de evidencias: En la ruta (parte superior izquierda del aula) daclic en Matemáticas. Se enlistarán las actividades de la unidad, da clic en Evidencia deaprendizaje: Álgebra de límites y continuidad.
Unidad 2. Límites y continuidad
Evidencias de aprendizaje: Álgebra de límites y continuidad
Da clic en Cierre de la Unidad. 2 de 2
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Matemáticas Administrativas Unidad 2. Límites y continuidad Evidencia de aprendizaje. Álgebra de límites y continuidad
Educación Superior Abierta y a Distancia • Ciencias Sociales y Administrativas 1
Evidencia de aprendizaje: Álgebra de límites y continuidad
Ejercicio 1 Cálculo de límites
Analiza cada una de las funciones que aparecen en la columna izquierda y compáralas con las respuestas de la columna derecha. Realiza los cálculos necesarios que te permitirán relacionar las columnas.
( )
()
()
()
()
1. -5.
2. 7.
3. 10
4.
5.
Nota. Debes incluir el procedimiento de cada uno de los planteamientos.
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Matemáticas Administrativas Unidad 2. Límites y continuidad Evidencia de aprendizaje. Álgebra de límites y continuidad
Educación Superior Abierta y a Distancia • Ciencias Sociales y Administrativas 2
Ejercicio 2 Rentabilidad con límites al infinito
Lee con atención el planteamiento del problema y elabora el procedimiento que se requiere.
Una cadena de hoteles desea construir un nuevo hotel en una zona donde recientemente se está
practicando ecoturismo, para lo cual calcula que la población turística dentro de años sigue la
siguiente función:
En cientos de turistas, si se ha determinado que se requiere que la población turística sea de al
menos 10000 personas en cualquier momento, determine si será rentable en algún momento
construir éste nuevo hotel:
Respuesta: ____________________
Solución:
Conclusión:
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