Limit of rational functions

The general expression of a limit is as follows:

limits1

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 Limits fr

We evaluate it first:

Fr2 limits

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

Fr3 limits

Simplified the expression we reevaluate the limit:

Fr4 limits

So:

Fr5 limits

Exercise 2: Calculate the value of the following limit Fr6 limits

We evaluate it first:

Fr7 limits

We factor the numerator of the expression:

Fr8 limits

Simplified the expression we reevaluate the limit:

Fr9 limits

So:

Limits fr10

Exercise 3: Calculate the value of the following limit Fr11 limits

We evaluate it first:

Limits fr12

Then we factor the numerator of the expression:

Limits fr13

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

Limits fr14

So:

Fr15 limits

Exercise 4: Calculate the value of the following limit Fr16 limits

We evaluate it first:

Limits fr17

We subtract the numerator from the expression:

Limits fr18

We divide the two fractions:

Limits fr19

We take common factor 3 from the numerator:

Fr20 limits

We take the common factor of the sign in the numerator:

Fr21 limits

We re-evaluate the limit:Fr22 limits