Lateral limits

The general expression of a limit is as follows:


Where lim is the abbreviated way of writing limit, f (x) is the function under study and x → a reads "when x tends to the value a in the function", that is, when the variable x takes values very close to the value a and L is the result of the limit.

When speaking of the limit of a function, it is understood that it is the study of its behavior, at a specific point, but if we apply the analysis (separately) between the numbers less than the point and greater than it, we are talking about lateral limits of a function.

If the limit on the left and right of that point do not have the same value, we could say that the limit does not exist, therefore, the lateral limits are a way to verify its existence. In general, we can express this theorem as follows:

lateral boundary1

Exercise 1: Calculate the limit lateral boundary2, being lateral boundary3.

We apply the limit of the function in each case, let's start on the left:

lateral boundary4

The limit on the right will be:

lateral boundary5

We can see that the limits are different, therefore this limit does not exist.

Exercise 2: Determine the limit lateral boundary6 being lateral boundary7.

We apply the limit of the function in each case; we start with the one on the left:

lateral boundary8

Now the limit on the right:

lateral boundary9

Like the lateral limits are the same, he limit exists.

Exercise 3: Determine if the following limit exists lateral boundary10.

We will simplify the limit so as not to have an indeterminacy and then solve what is inside the root:

lateral boundary11

As we have already reduced the expression we apply the lateral limits, remembering the absolute value:

lateral boundary12

Now the limit on the right:

lateral boundary13

Since the lateral limits are not equal, this limit does not exist.