LÍMITES al INFINITO 📈 Cómo Calcular Límites
Summary
TLDREn este vídeo tutorial, Susi enseña cómo calcular los límites cuando x tiende al infinito. Explica que al substituir x por infinito en una función, el término de mayor grado domina el resultado. Detalla cómo operar con infinitos, señalando que un número dividido entre infinito da cero, mientras que infinito dividido entre un número sigue siendo infinito. También menciona indeterminaciones como infinito dividido por infinito o infinito menos infinito, y promete explicárselos en futuras videos. Finalmente, invita a suscriptores y compartidores para recibir más contenido educativo.
Takeaways
- 🔢 Para calcular límites cuando x tiende a infinito, se sustituye x por infinito en la función.
- 🌌 Multiplicar un número por infinito resulta en infinito, y elevar infinito a cualquier potencia también es infinito.
- 📉 Al dividir un número por infinito, el resultado es cero, ya que se considera que infinito entre un número es casi nada.
- 📈 En fracciones, si el denominador tiende a infinito, el valor de la fracción tiende a cero.
- 🔄 Si el numerador y el denominador de una fracción tienden ambos a infinito, se puede llegar a una indeterminación.
- 🔄 Identificar indeterminaciones es crucial para resolver límites cuando x tiende a infinito.
- 📚 Se debe entender que el grado más alto de x en una fracción o polinomio es el que domina el comportamiento del límite.
- ⚠️ Cuídate de las raíces al calcular límites, ya que elevar a una fracción cambia el exponente.
- 🔄 En casos de indeterminaciones como infinito entre infinito o infinito menos infinito, se requiere un tratamiento especializado.
- 📘 Recordar que un número dividido entre infinito es cero y que infinito dividido entre un número es infinito, es fundamental para operar con infinitos.
Q & A
¿Qué tipo de límites se discuten en el video?
-Se discuten los límites cuando x tiende a infinito o a menos infinito.
¿Cómo se sugiere calcular los límites cuando x tiende a infinito?
-Se sugiere sustituir el infinito o el valor que x tiende por x en la función.
¿Qué pasa con la operación de infinito más infinito?
-La suma de infinitos da como resultado otro infinito.
¿Cómo afecta un número finito a un número infinito en una operación?
-Un número finito no afecta la magnitud de un número infinito; por ejemplo, infinito más seis sigue siendo infinito.
¿Cuál es el papel de los términos con mayor grado en una fracción cuando se evalúa el límite?
-Los términos con mayor grado en una fracción suelen dominar el resultado del límite.
¿Qué ocurre cuando se divide un número entre infinito?
-Cuando se divide un número entre infinito, el resultado tiende a cero.
¿Cómo se maneja la raíz de un número infinito?
-La raíz de un número infinito sigue siendo infinito.
¿Qué se debe tener en cuenta cuando hay exponentes fraccionarios en el límite?
-Cuando hay exponentes fraccionarios, el exponente interior se convierte en el numerador y el índice de la raíz se convierte en el denominador.
¿Qué es la indeterminación en los límites?
-La indeterminación es un resultado no determinado que se obtiene al evaluar un límite, como infinito dividido por infinito.
¿Cómo se sugiere aprender sobre los límites y las indeterminaciones?
-Se sugiere entender el concepto en lugar de memorizar, ya que la memoria puede fallar en momentos de estrés.
¿Cuáles son las reglas para recordar cuando se maneja infinito en una operación?
-Se mencionan reglas como 'un número entre infinito es cero' y 'infinito entre un número es infinito'.
Outlines
📚 Introducción a los límites al infinito
Susi da la bienvenida al canal e introduce el tema del video: cómo calcular límites cuando \( x \) tiende a infinito o a menos infinito. Explica que para resolver este tipo de límites rápidamente, se sustituye \( x \) por infinito en la función, y detalla cómo operar con números infinitos en expresiones polinómicas. El énfasis está en identificar el término de mayor grado de \( x \), ya que este domina el comportamiento del límite. También hace una demostración con ejemplos, mostrando que incluso cuando se suman o restan constantes pequeñas, el resultado sigue siendo infinito.
✍️ Operando con infinitos en fracciones
Susi enseña cómo calcular límites al infinito cuando las funciones involucran fracciones. Explica que al sustituir \( x \) por infinito, una fracción con un numerador finito y un denominador infinito tiende a cero, utilizando una analogía de dividir una torta entre infinitas personas. También describe cómo fracciones con infinitos en el numerador y denominador pueden llevar a indeterminaciones, las cuales se abordan en otro video. El enfoque está en la comprensión de conceptos, como la división entre números grandes y la importancia de identificar indeterminaciones.
🔍 Indeterminaciones y conclusiones
Se continúa explicando las indeterminaciones comunes al calcular límites al infinito, como infinito entre infinito e infinito menos infinito. Susi explica que estas indeterminaciones requieren un tratamiento especial, que se cubrirá en futuros videos. Al final, resume algunas reglas claves, como que un número dividido por infinito da cero y que infinito dividido por un número finito sigue siendo infinito. Cierra el video animando a los espectadores a suscribirse al canal, seguirla en Instagram y compartir el contenido.
Mindmap
Keywords
💡Límite
💡Infinito
💡Grado de la variable
💡Sustitución
💡Polinomio
💡Indeterminación
💡Fracción
💡Raíz
💡Exponente
💡Numerador y denominador
Highlights
Introducción a los límites cuando x tiende a infinito, y la importancia de este concepto en las matemáticas.
Explicación clara de cómo calcular límites cuando x tiende a infinito al sustituir infinito en la función.
Ejemplo práctico: sustitución de infinito en una función polinómica y análisis de cómo los términos dominantes afectan el resultado.
Explicación de cómo multiplicar y operar con el concepto de infinito, incluyendo la propiedad de que cualquier número multiplicado por infinito sigue siendo infinito.
Demostración de cómo el término de mayor grado domina en un polinomio al calcular límites cuando x tiende a infinito.
Análisis de los límites con potencias negativas y cómo afecta el signo de infinito al resultado.
Explicación de las raíces con exponentes fraccionarios y cómo modifican el comportamiento de las funciones al calcular límites.
Discusión sobre la importancia de identificar correctamente los grados de los términos en polinomios para calcular límites de manera eficiente.
Introducción al cálculo de límites con fracciones y cómo el concepto de dividir entre infinito resulta en cero.
Ejemplo práctico de cómo una fracción con infinito en el denominador tiende a cero.
Demostración de cómo dividir infinito entre un número finito sigue resultando en infinito.
Explicación del concepto de indeterminación y cómo ocurre en algunos límites cuando x tiende a infinito.
Reconocimiento de indeterminaciones típicas, como infinito dividido por infinito e infinito menos infinito.
Consejo sobre cómo recordar ciertos patrones al operar con infinito, como que dividir un número entre infinito da cero.
Cierre del video con un resumen de las indeterminaciones y recordatorio sobre los conceptos clave relacionados con los límites y el infinito.
Transcripts
Hello everyone, I'm Susi and welcome to my channel.
In this video we are going to learn to find limits when x tends to infinity,
so let's get to it.
Limits to infinity are those whose x tends to infinity or to minus infinity, as in this case.
We are going to learn to calculate this type of limits. Well, what we have to do to calculate them quickly
is to substitute the infinity or the value that the x tends to, which in this case is going to be infinity, in our function.
That is, every time it is equal to an x, I am going to substitute that x by infinity.
3 times x squared, well, infinity squared plus 5 times x, 5 times infinity minus 6.
And here, attentive to infinity, infinity is an immense number, huge, great, an infinite number, okay?
So how do we operate with this type of quantities? Well, thinking.
If I have infinity squared, it is another infinity bigger, but it is still infinity.
And 3 times infinity is still infinity. If I multiply 3 times something infinity, it is still infinity, very infinity.
Plus, that is, this would give me infinity squared, infinity, and 3 times infinity, it is still a very large number, it is still infinity.
Plus 5 times infinity, 5 times infinity, it is still infinity, okay?
5 times a very large number, well, it is still a very large number.
Minus 6, if here I have a very large number, an infinite plus another very large number, plus another infinite.
I have a very large number, so I add another very large number, it is still infinity, and if that infinity subtracts 6?
You can not notice, it is still infinity. Okay? Then this gives me infinity, okay?
So, in the end, when it comes to operating with infinites,
when we have x, the x of greater degree is a bit the one that rules, because
adding a smaller infinite, or a number, or subtracting it, is not going to influence the larger infinite.
So, in the end, what is done? When we have a polynomial function,
what we do is simply take the greater degree, as in this case, we are going to do it,
and we only substitute there, okay? There is no need to do all this development.
If I only take the greater degree of the x, I can do it directly,
that is, I only substitute the infinite in the x with greater degree, okay?
Infinite to the cube is infinite again, and two times that infinite, infinite.
Let's see here, what is the x that has greater degree, this one?
Well, in this case, we have to substitute it for minus infinite.
Well, two times minus infinite to the square, okay?
Well, a very large negative number, raised to the square,
if it is raised to the square, it will be minus infinite times minus infinite,
minus times minus plus, and it is infinite, okay?
That is, it gives a very large number, and two times plus infinite, it will be again plus infinite, okay?
What is going to give us here the same thing?
We have a polynomial, we take the greater degree, and we substitute the minus infinite by the x.
Here, since it is minus infinite to the cube, being negative to the cube,
minus times minus plus, times minus, minus.
Minus infinite to the cube is minus infinite, and five times minus infinite, minus infinite, okay?
And be careful here when you have roots.
Apparently, the greater degree is x raised to five, minus x raised to four,
but be careful because the root modifies the exponent. Why?
Because if we pass this to a fractional exponent,
remember that a root is passed to an exponent as a fraction.
The exponent of the interior is put in the numerator of that fraction, which in this case is five,
and in the denominator we put the index number, which in the case of a square root,
you already know that it is two, even if it is not put.
Therefore, the exponent of this, of the square root of x raised to five is actually five halfs,
which is smaller than four. Five halfs given with something, and four is less than four.
Therefore, here the greater exponent, be careful, which is this, minus x raised to four, okay?
Well, that's where we have to substitute.
Minus x raised to four is going to be minus infinity raised to four.
Be careful with this negative. The negative does not affect the x.
If I put parentheses, yes, but the negative outside.
Therefore, this minus, we do first the infinity raised to four.
Infinity raised to four is infinite, but as I have the negative in front,
it is actually minus infinity.
Let's see how these limits would be calculated if what we have are fractions, okay?
You see? In these fractions, well, and here in this case we will see with this root what happens.
Well, as we have said, we substitute the x that appears to us for infinity.
Okay? So, one divided by x squared, infinity squared, minus one.
One divided by infinity squared, a very large number squared,
it is still an even larger number, but it is still a large number, that is, an infinity, minus one.
Here, one up, and if there is infinity, I remove one,
well, it is hardly modified, it is still infinity.
And here, be careful, a number divided by infinity is going to give zero, why?
Let's think about it, if I have one, a cake, and I have to divide it between infinite people,
how much cake does it cost you? Nothing, nothing, you hardly take cake.
So, almost nothing is a zero, okay?
So, to help you agree, that, a number divided by infinity is going to give zero, okay?
And that is the result of this limit.
We are going to do this, the same, I substitute x tends to infinity,
well, I substitute the x for infinity, infinity squared, minus one divided by two.
Infinity squared is infinity.
If there is infinity, I remove one, the same as before, it is still infinity divided by two.
And here, backwards, when I have infinity between a number, let's think,
I have to repeat, I don't know, all the stars in the universe,
between two people, how many stars does each person touch?
There are also infinities, okay?
Since I have to divide an immense amount between only two people, each person continues to carry an immense amount,
that is, each person carries infinity.
So, remember this too, infinity between a number, infinity, okay?
If you denote it like that, you don't have to agree, okay?
Then, if you want, I'll write you a little bit of a joke here,
and with that, if you want to learn it from memory,
but you already know that memory, when you are nervous, when you have an exam and you are limited in time,
memory often fails, and on top of that, if you haven't slept, even more.
So try to understand things, which will be much better.
Limit of when x has infinity.
In this fraction, I substitute the x that appears by infinity,
infinity squared plus three times x, that is, three times infinity, divided by infinity minus five, okay?
Here, just like up here, I have infinity squared, that is, infinity plus another infinity,
I'm going to have infinity up here, and down here, infinity.
If I take away five from infinity, I still have infinity, and here, be careful, be careful with this.
This is called indetermination, okay?
It is a value that is not determined from there, indetermination, okay?
So, there are some indeterminations that you have to know how to identify,
and when they are available later, you can click on the box and there I show you how to calculate indeterminations.
This is a typical termination for this type of limit to infinity.
Let's see here, let's also substitute infinity, that is, if we can't find the indeterminations, sorry,
we leave it here, I know that some of you teach you to do this process first,
and when you get to indetermination, you just put indetermination.
If you have to learn how to solve it, as I said, I will explain it in another video, in more detail.
That's it, we substitute for infinity each x, minus five times x times five times infinity divided by three minus two times infinity.
If I continue here, here, infinity squared times two minus five times infinity,
this is going to be infinity minus another infinity, divided by three minus two times infinity,
I will have infinity, infinity, that is, infinity divided by three, infinity, minus two times infinity, infinity.
This is also another indetermination, okay?
And here the same thing, I'm going to substitute infinity squared, root of infinity squared, minus infinity, minus infinity.
Infinity squared minus infinity is still infinity, because it wins this, infinity squared, if I remove another infinity, it is still infinity.
The root of infinity is infinity, infinity minus infinity, the same as before, another indetermination, okay?
From here, things that you have wanted to put, and typical indeterminations of these limits,
infinity divided by infinity, infinity minus infinity, which we will learn there in another video,
more in detail, and things to keep in mind, that number between infinity is equal to zero, infinity between a number is equal to infinity, okay?
The little thing I promised you before, if we remember this, we have no problem when it comes to operating with infinities.
And so far today's video, if you liked the video, give it a like and share it.
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Have a good day and see you in the next video.
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