In this post I’m going to explain the limits of functions. We’ll look at the most basic things you need to know to understand limits: what function limits are and how they are resolved (in general).

I will start by giving you a definition of limit so that you understand the concept and then we will continue with the resolution of simple limits so that you understand everything perfectly. This is the basis from which you can learn to solve more complex limits with indeterminations.

I’ll explain to you what the lateral limits are, too.

## What is the limit of a function?

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It is represented as follows:

Which means, as I have just told you, that when X tends to the point Xo, the value of the function approaches L, therefore, the limit of that function when X tends to Xo is L. Graphically it would look like this:

If you notice, as we approach the Xo value on the x-axis, on the y-axis, the value of the function will approach the L value.

x can tend to any value, from less infinite to more infinite (both included) and the limit of a function can also be from less infinite to infinite (both included).

Do not confuse the limit of a function with the value of a function in point, which is the value of the function right at that point. Be very careful because they may not match (see below).

Let’s look at it with an example.

What is the limit of the next function:

when x tends to -1?

The limit of the function when x tends to -1 is written:

So that you understand how the value of the function will approach a certain value, while x will tend to -1, we will see what the value of the function is for the points that are close to -1 and each time we will get closer to -1.

First we’re going to get closer and closer to -1 on the left to see what happens.

When x=-1.3, the value of the function is:

When x=-1,2, the value of the function is:

When x=-1.1, the value of the function is:

If you notice, as we get closer to -1, the value of the function gets closer to 0.

We’re going to do the same thing now, but approaching 1 on the right.

When x=-0,7, the value of the function is:

When x=-0,8, the value of the function is:

When x=-0,9, the value of the function is:

As you can see, as we get closer to x=-1 from the right, the function gets closer and closer to 0.

If we look at it graphically, we see how the graph of the function approaches point 0 on the y-axis, when the values of x are approaching point -1 on the x-axis:

Therefore, the limit of the function when x tends to -1 is equal to 0:

To solve a limit it is not necessary to perform this procedure that we have just done. I only did it so that you could see how little by little the value of the function is approaching a point.

Solving the limit of that function is much easier and that is what I will explain in the following section.

## How to solve the limit of a function

In cases where the domain of the function is all R (the function is continuous in all R), such as polynomials, the limit of the function in a point will be calculated the same as the value of the function in that point, i.e. replacing the value by x.

Let’s solve the limit of the previous function when x tends to -1:

To solve it, we have to replace the x with -1 and operate:

And we get the result of the limit which is 0.

Notice when we replace the value to which the x tends by the x, the limit disappears.

In this case, the limit of the function when x tends to -1, and the value of the function at -1 coincide, but this does not have to be the case.

In functions that are not continuous (the domain is not all R), there are points where the limit has a value and yet the function at that point does not exist or the value of the function has a different value.

For example, in the next function:

The limit when x tends to -1 is equal to 0, but however the value of the function in x=-1 is equal to 2:

Or the case of this other function:

Let’s see what happens if we calculate the limit of the function when x tends to 1:

We replace the x with the 1:

And we can’t come up with a solution, because a number between zero is an indeterminacy. In the Limits Course I will explain in detail the types of indeterminations and how they are resolved.

## Lateral limits

We have seen before that the limit of a function is the value to which that function approaches when x tends to a certain point, both on the left and on the right.

However, you can calculate the limit of a function when approaching only from the left or when approaching only from the right. They’re called lateral boundaries.

The value of the lateral limits of a function may or may not coincide.

For the limit of a function to exist at a point, the value of the lateral limits must coincide and that will be the limit value at that point.

If the lateral limits do not coincide, then the limit does not exist.

Let’s look at it with an example so you can see it more clearly.

We have the following function:

Let’s calculate how much the limit of the function is worth when x tends to 4.

If we approach x=4 from the left, the limit is 3:

When the lateral limit is to the left, a minus sign is added as an exponent to the value of x to which the limit extends.

If we approach x=4 from the right, the limit is equal to 5:

When the lateral limit is to the right, a plus sign is added as an exponent to the value of x to which the limit extends.

The lateral limits in this case do not coincide, therefore the limit of the function when x tends to 4 (on both sides) does not exist:

The «does not exist» symbol is a capital E in reverse.