Indeterminations zero for infinity, infinity minus infinity and 1 raised to infinity

Now I will explain how to calculate limits with indeterminations zero for infinity, infinity minus infinity and 1 raised to infinity. We will see it in detail while with step-by-step exercises resolved.

The resolution of these indeterminations are chained with other types of indeterminations such as infinity among infinity, so you must also know how to resolve that indetermination (you have it explained in the Course of Limits, along with all indeterminations).

Calculation of limits with zero by infinity indetermination

Any number multiplied by zero is zero, but in turn, any number multiplied by infinity is infinite. So What is zero for infinity? Is it zero? Is it infinity?

Well, it’s neither one thing nor the other. Therefore, zero for infinity is another indeterminacy.

infinity by zero

The procedure for solving limits with zero indetermination by infinity is:

  1. To reach zero indetermination by infinite by substituting the x for the number you shop for
  2. Operate within the function to eliminate indeterminacy
  3. Solve the infinite indeterminacy between the infinite that is left to us

Let’s solve an example of a limit with zero indetermination by infinity:

zero by infinity

First of all, we replace the x with infinity and we come to the conclusion that we are facing a zero indeterminacy for infinity:

infinite by infinite

To solve the indeterminations of zero by infinity, what must be done first of all is to operate within the limit. In this case, we can multiply the fraction by the parenthesis and it remains:

infinity raised to infinity

We replace the x with infinity again and arrive at another indeterminacy, this time of infinity between infinity:

infinite - infinite

To solve the infinite indeterminations between infinity, we stay in the numerator and denominator with the term of the highest degree and then we operate:

cuanto es infinito por cero

We replace the x with infinity and arrive at the final solution of the limit, which is infinity:

infinite among infinite

We are going to solve another limit exercise with this kind of indetermination.

infinito/infinito

We begin by substituting the x for infinity and we reach the point where it has a zero indeterminacy for infinity:

cuanto es 0 por infinito

We operate in the function by multiplying the fraction by the root and it remains:

1 entre infinito

We replace the x with infinity and arrive at the result of the infinite indeterminacy between infinity:

cuanto es cero por infinito

We keep the highest grade terms of the numerator and denominator and solve the root that is left in the numerator:

0 by infinity

We can eliminate the x of both parts of the fraction and therefore, we reach the final result:

infinite for zero as much as

Limit calculation with infinite indetermination minus infinity

Let me ask you a question: Infinity minus infinity is zero?

A lot of people would say yes, but not really. Although it is an abstract concept, there are many “sizes” of infinity, but we do not know. Since we do not know whether infinity is equal, infinity minus infinity is another indeterminacy:

limite infinito elevado a infinito

The procedure for resolving boundaries with infinite indeterminacy minus infinite is as follows:

  1. To reach infinite indeterminacy less infinite by substituting the x for the number you shop for
  2. Multiply and divide by the function conjugate
  3. Operate on the numerator of the resulting fraction to simplify term
  4. Solve the infinite indeterminacy between the infinite that is left to us

 

 

Let’s solve an example of a limit with infinite indeterminacy less infinite less infinite slower:

indeterminacion infinito menos infinito

First, we replace the x with infinity, which brings us to infinite indeterminacy less infinite:

indeterminacion 1 elevado a menos infinito

To solve limits with infinite indeterminacy, less infinite, we must multiply and divide them by the conjugation of function:

limit infinite by infinite

We operate in the fraction numerator, where we have a difference of squares, since we had a multiplication of sum per difference:

indeterminacion infinito por infinito

We solve the squares and group similar terms in the numerator:

indeterminacion infinito por cero

We replace the x with infinity. We arrive at another indeterminacy, which is now the one of infinity among infinity:

indeterminacion cero por infinito

We keep the highest grade term in both the numerator and the denominator, solve the root in and group terms in the denominator:

indeterminacion 1 elevado a infinite ejercicios resueltos

Finally, we eliminate the x and y to reach the limit result:

1/infinito

Calculation of limits with indeterminacy 1 raised to infinity

If I asked you how much 1 is elevated to infinity, would you say the result is 1? For you would not be right, for 1 elevated to infinity is another indeterminacy:

zero for infinity as much as

To calculate the limits with indeterminacy 1 raised to infinity, calculate using this formula:

infinity between zero

Where the number to which the x tends can be any number or it can be more or less infinite:

indeterminacion infinito elevado a infinito

indeterminacion 0 por infinito

We will follow the following procedure to resolve limits with indeterminacy 1 raised to infinity:

  1. To reach indeterminacy 1 raised to infinity x by the number you are tending to
    1. Solve, if necessary, the infinite indetermination between infinity of the limit of the function that forms the basis of power, to show that its result is 1
  2. Apply the formula to solve indeterminations 1 raised to infinity
  3. Perform operations on the function, within the limit
  4. Solve the infinite indeterminacy between the infinite that is left to us

 

 

Let’s solve step by step an example of a limit with indeterminacy 1 raised to infinity:

infinity raised to less infinity

We began to solve this limit by substituting the x for infinity and arrived at this result:

1 over infinity

This indeterminacy is not 1 elevated to infinity, but in order to apply the above formula, we must demonstrate that the infinity in infinity that is in parentheses is equal to 1.

To do this we solve the limit of the function that forms the basis of the power, that is to say:

number multiplied by infinity

We replace the x with infinity and reach the infinite indeterminacy between infinity:

zero between infinity

We keep the largest term of the numerator and denominator and operate, arriving at the result, which is equal to 1:

e elevado a menos infinito

Knowing that the limit of the function, which forms the basis of the power, when x tends to infinity is 1, now we can say that this limit results in indeterminacy 1 raised to infinity:

indeterminacion 1 elevado a infinito

Therefore, we apply the previous formula and we are left with it:

infinite over infinite

Once the formula is applied, we have to do operations. First, within the parenthesis, we subtract by reducing the common denominator and group terms in the numerator:

1^infinito

We now remove the parenthesis by multiplying it by the term before it:

indeterminaciones 0 por infinito

When we can no longer operate, we replace the x with infinity and reach the infinite indeterminacy between infinity:

indeterminaciones infinito menos infinito

To resolve this indeterminacy, we leave the term of highest degree and operate:

limit infinite by zero

Finally, we replace the x by infinite again, which is raised to less infinite by “e” than by properties of the powers, lower the denominator. “e” raised to infinity equals infinity and 1 split by infinity equals zero:

indeterminacion 0 por infinito ejercicios resueltos

Calculation of limits with infinity raised to infinity

Let’s see how to solve the limits in which once you replace the x with infinity, you have as a result infinity raised to infinity.

To begin with, infinity elevated to infinity is not an indeterminacy. Infinity elevated to infinity equals infinity:

infinity between infinity raised to infinity

Therefore, to solve this type of limit, we only have to replace the x by infinite and the result is obtained directly.

For example:

limites infinito elevado a infinito

We replace the x by infinite, which remains infinite to infinite, whose result is infinite:

limit 0 by infinity