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While studying calculus or other branches of mathematics we may need to find the limits of a function, a sequence, or an expression, and in doing so we stumble on a situation where we cannot determine the limits, in this article we will learn about the different indeterminate forms and how to work around them in order to find the limits we are looking for.

## Indeterminate Forms

We call an indeterminate form, when computing limits the case when we get an expression that we cannot determine the limit. In total there is seven indeterminate forms, here they are:

Here are some examples to illustrate each of these indeterminate cases:

Indeterminate form

Indeterminate form

Indeterminate form

Indeterminate form

Indeterminate form

Indeterminate form

Indeterminate form

## L’Hôpital’s rule and how to solve indeterminate forms

L’Hôpital’s rule is a method used to evaluate limits when we have the case of a quotient of two functions giving us the indeterminate form of the type or .

The L’Hôpital rule states the following:

**Theorem: L’Hôpital’s Rule:**

To determine the limit of

where is a real number or infinity, and if we have one of the following cases:

Then we calculate the limit of the derivatives of the quotient of and , i.e.,

**Examples:**

**Case of** :

**Case of **:

In this case, after we get the derivatives of the quotient, we still get the indeterminate form of the type so we apply L’Hôpital’s Rule again, and therefore we get:

For other Indeterminate forms, we have to do some transformation on the expression to bring it to one of the two forms that L’Hôpital’s rule solves. Let’s see some examples of how to do that!!!

**L’Hôpital’s rule with the form **:

Let’s compute

Here we have the indeterminate form , to use L’Hôpital’s rule we re-write the expression as follow:

Now by computing the limit we have the form , therefore we can apply L’Hôpital’s rule and we get:

**L’Hôpital’s rule with the form **:

Let’s compute

This limit gives us the form , to apply the L’Hôpital’s rule we need to take a few steps as follow:

Let’s be:

By applying the natural logarithm, we get:

And now we compute the limit:

And since we know that:

Therefore, we can write the limit as:

And from what we got before; we can solve the problem as follow:

**L’Hôpital’s rule with the form **:

Let’s compute

This limit gives us the form , to apply the L’Hôpital’s rule we need to re-write the expression, in this case, all we need to do is combine the two fractions as follow:

Now the limit of the expression gives us the form . Now by applying the L’Hôpital’s rule twice (because we get the indeterminate form after the first time) we get:

**L’Hôpital’s rule with the form **:

Let’s compute

This limit gives us the form , to avoid it and be able to apply L’Hôpital’s rule we need to re-write the expression as follow:

Let

Then

Using L’Hôpital’s rule we get:

And therefore, we get:

**L’Hôpital’s rule with the form **:

Let’s compute

This limit gives us the indeterminate form , to use the L’Hôpital’s rule we need to re-write the expression as follow:

Now we calculate the limit of the exponent using L’Hôpital’s rule:

Therefore,

## Limits of a composite function

**Theorem:**

Let , and represent real numbers or or , and let , , and be functions that verify .

If the limit of the function when tends to is , and the limit of the function when tend to is then the limit of the function when tends to is .

Meaning: if and if then

**Example:**

Let’s consider the function defined on the domain as

and we want to determine the limit of the function when tends to , i.e.,

We notice that the function is a composite of two functions, precisely is a composite of the functions and in this order (), where

and

Since

And

Therefore

## Limits with comparisons

**Theorem 1:**

Suppose , , and three functions, and a real number; if we have and and if for big enough we have then .

**Example:**

Let’s consider the function defined on as

We know that for every from , we have

And therefore, for every in , we have

And since we conclude that

**Theorem 2: **

Suppose and two functions and a real number; if we have , and if for big enough we have then .

**Theorem 3:**

Suppose and two functions and a real number; if we have , and if for big enough we have then .

**Remarque:** these three theorems can be extended to the two cases for the limit when tends to or a real number.

**Example:**

Let’s consider the function defined on as

We know that for every from , we have , and then for every from , we have

Therefore:

Since

Then

And since

Then

## Conclusion

In this article, we discovered the different indeterminate forms and how to avoid them and calculate the limits using L’Hôpital’s rule, with examples of the various cases. Also, we learned about how to determine the limits of composite function and how to determine limits with comparison. Don’t miss the previous articles about the idea of limits, their properties, and the arithmetic operations on them.

Also, if you want to learn more fun subjects, check the post about Functions and some of their properties, or the one about How to solve polynomial equations of first, second, and third degrees!!!!!

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