# Limit of rational functions

The general expression of a limit is as follows:

Where lim is the abbreviated way of writing limit, x → a is read "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.

In the study of the limits of rational functions, we must be careful with the indeterminacies that are presented to us and have knowledge of algebra, such as factorization, empowerment, among others. There is no rule to solve a limit, however, the first thing we will do when facing a problem will be to evaluate it, then find a way to simplify it and reevaluate it in order to obtain its final value. Let's see some exercises:

Exercise 1: Calculate the value of the following limit

We evaluate it first:

We find an indeterminacy, in order to solve this limit we must factor the numerator of the expression:

Simplified the expression we reevaluate the limit:

So:

Exercise 2: Calculate the value of the following limit

We evaluate it first:

We factor the numerator of the expression:

Simplified the expression we reevaluate the limit:

So:

Exercise 3: Calculate the value of the following limit

We evaluate it first:

Then we factor the numerator of the expression:

Since we have simplified the expression, we evaluate the limit:

So:

Exercise 4: Calculate the value of the following limit

We evaluate it first:

We subtract the numerator from the expression:

We divide the two fractions:

We take common factor 3 from the numerator: