Resolution of limits with indetermination of a number between zero. Exercises resolved.

In this lesson I will explain how to solve the limits with indeterminacy a number between zero or indeterminacy k/0. I will explain it to you at the same time that we are solving exercises, so that you know all the steps in detail.

To understand well the limits with indeterminacy number divided by zero, you have to understand well how the lateral limits work, since we will use them to solve this type of indeterminacy.

Result of dividing a number by zero

What is the result of dividing a number by zero?

limit when x tends to 0

Many people think that when a number is divided by zero, the result is infinite, but that’s a mistake:

limits when x tends to 0

To begin with, a number between zero is an indeterminacy, that is, it has no solution:

limits that tend to zero

However, another very different thing is to calculate the limit of a function with an indeterminate number between zero.

The result of this limit may be “more infinite”, “less infinite” or it may not exist. But we are talking about the limit, that is to say, the value to which the function approaches when x has the number. Right at the point where the function is worth a number between zero, the function does not exist, because that value does not exist.

Therefore, what you have to stay with is that a number between zero is an indeterminacy, but that the limit with an indeterminacy number between zero can be resolved and may or may not exist, which is what we will see in the next section.
How to solve the limits with indeterminacy number between zero

As I have told you before, the limit with an indeterminate number divided by zero may have a solution, which will always be “more infinite” or “less infinite” or the limit may not exist.

what happens when a limit tends to zero

How do we solve the limits with indeterminacy number between zero?

They are solved by calculating the lateral limits.

The result of the lateral limits can be “more infinite” or “less infinite”. If the zero is positive the result is “more infinite” and if the zero is negative the result is “less infinite”:

limit when x tends to 0 examples

limit x tends to 0

I’m sure you’re wondering: What do you mean, zero positive and zero negative? zero is not in the middle of positive and negative numbers and so it’s neither positive nor negative? how’s that?

I’ll explain this in detail below, don’t worry, but I want you to keep the results that can have the lateral limits for now.

Continuing with the resolution of limits with the indeterminacy number between zero, if both the limit on the right and the limit on the left coincide, then the limit will have a solution, which will be the solution of the two lateral limits.

If the lateral limits do not coincide, then there is no limit that we are calculating.

Resolved exercises of limits with indeterminacy number between zero

Let’s look at it with some exercises worked out to make it clearer.

Exercise resolved 1

Resolve the following limit:

limit tends to zero

First, as with all limits, we replace the x with the 2 and we arrive at the result of a number between zero, which you know is an indeterminacy:

un numero entre cero

Therefore, we have to calculate the lateral limits, that is, the limit of the function when x tends to 2 on the right and the limit when x tends to 2 on the left.

We will see if the result of both lateral limits coincide and the limit exists, or they do not coincide and the limit does not exist.

We start with the limit when x tends to 2 from the right:

indeterminacion k/0

To calculate the limit when x tends to 2 on the right, we consider that 2 on the right is a value very close to 2, but that is a little more than 2. To see it more concretely, I usually add 0.0001:

zero between a number

Then, we replace this new value in the function:

limits when x tends to zero

In terms of numerator, we’d still have a result of a little more than 6, but it’s still a number and we don’t care, so we keep putting 6.

However, the key is in the denominator. When you substitute 2.0001, the result is no longer zero, but a value that is very close to 0, but is positive. We will call this value +0 (zero positive):

limites k/0

The positive zero is a value very close to 0, to the right, that is, it is positive and does not reach zero.

This means that when we approach 2 on the right, we are dividing by a value very close to zero, which is positive, the result of which tends to be “more infinite”, so that the result of the limit is no longer indeterminate, but is “more infinite”:

limit 0 over number

Therefore, the limit of the function when x tends to 2 on the right is equal to more infinite.

Let’s calculate now when the limit when x tends to 2 from the left:

limit of a number above zero

As before, to calculate the limit when x tends to 2 on the left, this time we consider that 2 on the left is a value very close to 2, but that is a little less than 2. In this case, to see it in a more concrete way, I usually subtract 0.0001:

limit when tending to zero

We replace the value of 2 by the left in the function:

limits when x tends to a number

In the numerator, we would have a result a little less than 6, but it is still a number and we don’t care, so we keep putting 6.

We look again at the denominator.

When you substitute 1.9999, the result is a value that is very close to 0, but is negative. We will call this value -0 (zero negative):

limits with indeterminacion

The negative zero is a value very close to 0, to the left, that is, it is negative and does not reach zero.

Therefore, when we approach 2 from the left, we are dividing by a value very close to zero, which is negative, the result of which tends to be “less infinite”, so that the result of the limit is no longer indeterminate, but is “less infinite”:

when the limit tends to 0

The limit of the function when x tends to 2 from the left is less than infinite.

In this case, the lateral limits do not coincide. Therefore, the limit of the function we were calculating has no solution, so there is no limit:

limit when x tends to a number

Let’s look at another example, which we will solve more quickly.

Exercise resolved 2

Resolve the following limit:

lim x tiende a 0

To solve this limit when x tends to 0, as in any other limit, we replace x with 0 and we reach the indeterminacy number between zero:

limit of x when it tends to 0

We must therefore calculate the lateral limits.

We start from the limit when x tends to zero on the right:

limits when x tends to 0

We consider that the zero on the right is a value very close to zero, so it is equivalent to adding 0.0001:

limit tends to 0

We replace the value of zero with the right in the function:

limits when tending to 0

The denominator is a positive zero:

limits x tends to 0

Therefore, when we divide by a value very close to zero, by its right, the limit is equal to infinity:

limit exercises when x tends to 0

We calculate the limit of the function when x tends to zero to the left:

lim cuando x tiende a 0

We consider that zero to the left is equivalent to subtracting 0.0001:

limits that tend to 0

We replace the value of zero with the left in the function:

limit solution

In this case, as the x is squared, although we had a negative number, it becomes positive, so the denominator result is also a positive zero:

when x tends to 0

And the limit of the function when x tends to zero to the left is also infinite:

limit of a function when x tends to 0

Both lateral limits coincide, whose result is infinite, therefore, there is the original limit that we were calculating, whose result is also infinite:

examples of limits when x tends to 0