Now I will explain **how to solve the infinite indeterminacy between infinite and infinite in the calculation of limits**.

Normally, to solve the limit, we only have to replace the x by the value it tends to.

However, there are times when we find that when replacing, the result is an indeterminacy and in that case, we must use the calculation method that corresponds in each case.

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## How to calculate limits with indeterminations of the infinite type between infinite and infinite. Exercises resolved.

This type of indeterminacy is found in the limits of when x tends to be infinite (or less infinite) of rational functions and when replacing it, we have ∞/∞.

In general they are have this shape:

When we have the indeterminacy of infinite limits between infinity, we have to remove terms from the numerator and denominator and leave only the highest degree terms up and down.

Once we have the highest grade terms, we can operate and eliminate the x’s that are repeated in both the numerator and denominator.

When operating, the indeterminacy disappears and we must replace the x again by the number it tends to reach the result.

We are going to see it step by step with several exercises solved as an example. We start with this limit:

All limits begin with the same resolution: substituting the x for the number it tends to, that is, it is not resolved knowing before beginning that it will be an indeterminacy. We don’t have to know that.

Once we have replaced the x and y in operation, then we can either arrive at a concrete result or come to the conclusion that it is an indeterminacy.

First, we replace the x with infinity and we are left with an indeterminacy:

To resolve this type of indeterminacy we leave only the term of the highest degree in the numerator and denominator:

Once we have left the term of higher degree, we operate and eliminate the factors that are repeated in both the numerator and the denominator. In this case, we can eliminate an x on each side:

And finally, we have replaced the x with infinity again, but this time, we no longer have any indeterminacy and the result is infinite among a number, which is equal to infinity.

Let’s see another example:

We start by substituting the x for infinity and then we’re left with it:

This time we have an infinite negative in the denominator, but it is still the same kind of indeterminacy.

We leave the term with the highest degree of numerator and denominator:

We operate by removing an x from the numerator and another from the denominator and replace the x with infinity again, arriving at the result of less infinity, as we are dividing by a negative number:

Here’s another example:

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

We leave the term of highest degree in the numerator and in the denominator:

We operate and can eliminate an x in both the numerator and the denominator. This time in the numerator we don’t have any x, so when it comes to substituting, the infinite is left only in the denominator and therefore, a number between infinite, gives us zero as a result:

Let’s end the infinite indeterminacy broken by the infinite with this example:

We replace the x by infinite and the infinite indeterminacy remains split by infinite:

We leave the highest grade term up and down:

When operating, we can eliminate x squared in both the numerator and denominator, so that the x’s disappear. Therefore, when it comes to replacing x with infinity again, as we do not have x, the result is the same number that we have left:

## Infinite indetermination between infinity with roots

When we meet radicals on the edge, it often causes confusion. We will see how to solve them if we have the indeterminacy of infinity between infinity and roots.

For example:

We replace the x with infinity and come to the conclusion that it is an indeterminacy:

We are left with the term of the highest degree, but now we must be careful with the radicals. In the numerator, the term with the highest degree is the first, since the second term is within another root and therefore the degree is lower.

In the denominator, the highest grade term is the root term.

We are therefore left with the same term above and below:

When operating, we are left with the function equal to 1, since both terms can be divided between them, and therefore, the limit of the function when x tends to infinity is equal to 1:

Let’s see another example:

We replace the x with infinity and we have the indeterminacy of infinity for infinity.

We leave the term of highest degree in the numerator and in the denominator:

And now we have to trade, but it’s not as clear how to trade to eliminate one of the x’s, as in the other examples we’ve seen.

In the first place, we pass the root to the fractional exponent:

And now, to operate with the x’s we consider them as a division of powers of the same base, where we keep the x’s and subtract the exponents:

And there is only one x left in the numerator, which when you substitute infinity, the result is infinite:

## Conclusion of the infinite indetermination between infinity

With the indeterminacy of infinity between infinity, the result will depend on the degree of the polynomials of the numerator and the denominator.

- If the degree of P(x) is less than the degree of Q(x), the result will be zero
- If the degree of P(x) is greater than the degree of Q(x) the result will be infinite or less infinite
- If the grade of P(x) is the same as the grade of Q(x) the result will be a number

In the solved examples that we have solved throughout the lesson, we have seen each of the cases at the same time.