Asymptotes of a function: how they are calculated. Exercises resolved step by step.

Here I will explain what an asymptote is and how to calculate the asymptotes of a function, as well as how many types of asymptotes there are. We will also apply what we have learned by solving some exercises on asymptote calculation.

What is an asymptote?

An asymptote is a line to which the graph of the function gets closer and closer, but which it never touches, although theoretically they are said to touch in infinity.

There are three types of asymptotes: horizontal asymptotes, vertical asymptotes and oblique asymptotes. Let’s see how to calculate each of them.

Horizontal asymptotes

Horizontal asymptotes are horizontal straight lines that the function never touches.

There will be a horizontal asymptote when the limit of the function when x tents to infinity is equal to a certain number k:

asintotasIn that case, in y=k, there will be a horizontal asymptote:

asintotic exercises

In the same way, when the limit of the function when x tends to be less than infinite is equal to a certain number, there will also be a horizontal asymptote:

asintotas de una funcion ejercicios resueltos

asintotas de una funcion

If each limit results in a different number, then that function has two horizontal asymptotes.

That number that results in the limit has to be a finite number and therefore can never be more or less infinite:

asintotas ejercicios

In the event that the above limits result in more or less infinite, there will be no horizontal asymptotes.

Vertical asymptotes

Vertical asymptotes are vertical straight lines to which the function gets closer and closer, but it will never touch.

There will be a vertical asymptote when the limit of the function when x tends to a number of as a result more or less infinite:

asintota horizontal

In that case, there will be a vertical asymptote at x=k:

asintota

And what is the k-number to calculate the limit for finding the vertical asymptote?

The number with which the vertical asymptotes are calculated is the number for which the domain of the function is not defined, i.e. the number that does not belong to the domain.

It may be that more than one number does not belong to the domain, so the function will have more than one vertical asymptote.

To calculate the vertical asymptotes we use the lateral limits, that it is not necessary for both lateral limits to have the same result for the vertical asymptote to exist, in contrast to what happens if we want to check if the limit of the function exists when x tends to a point.

For example, if at a particular point, one side boundary gives more infinity and the other less infinity, there will be a vertical asymptote, but there will be no limit to the function at that point.

If the mastery of the function is all R, there will be no vertical asymptotes.

Oblique asymptotes

Oblique asymptotes are only calculated if there are no horizontal asymptotes.

Like the other two types of asymptotes, oblique asymptotes are oblique straight lines, to which the function gets closer and closer, but never touches.

As it is an oblique line, it has this shape:

how to calculate asintotas

And it is about calculating the coefficients m and n to find the equation of the line.

To calculate the coefficient m we use the following formula:

asintotic exercises resolved

For the oblique asymptote to exist, m cannot be equal to zero, since if m=0, the asymptote would be horizontal:

calcular asintotas

The coefficient m cannot be infinite either, because otherwise the asymptote would be vertical:

asintotas exercises solved step by step

The coefficient n is calculated with the following formula:

asintotas oblique exercises resolved

And finally, once the values of the coefficients m and n have been obtained, we would have the equation of the line that defines the oblique asymptote:

exercises asintotas

Resolved asymptote calculation exercises

We’re going to work out some exercises to apply the whole theory I just told you about.

Calculate the asymptotes of the next function:

find the asynths of the following functions

Calculation of horizontal asymptotes

We calculate the limit of the function when x tends to infinity:

asintotic resolved exercises

We replace the x with infinity and arrive at the result of the infinite indeterminacy divided by infinity:

asintotas verticales y horizontales ejercicios resueltos

We are left with the term of the highest degree and arrive at the result of the limit:

asintotas horizontal exercises resolved

Therefore, at y=1 there is a vertical asymptote:

asintota vertical

We also calculate the limit of when x tends to be less infinite:

asintotas ejercicios resueltos

We replace the x with less infinite and we reach indeterminacy:

asintotas verticales ejercicios resueltos

We solve it and get to the same result as before:

calculo de asintotas

Therefore, we only have a horizontal asymptote that is at y=1:

asintotas oblique exercises

Calculation of vertical asymptotes

To calculate the vertical asymptotes, we must first know what is the domain of the function. The mastery of function is:

examples of asynths

The numbers that do not belong to the domain are the ones we have to calculate the limit with, that is, 1 and -1.

We begin by calculating the limit of the function when x tends to 1:

asintota oblique exercises

We replace the x with 1 and arrive at the indeterminacy of a number between zero, which we do not know if it is infinite or less infinite:

asyntotas

The limits of this type of indetermination are resolved using lateral limits.

We begin by calculating the limit of the function when x tends to be 1 on the left, whose result is less infinite:

asintotas verticales horizontales y oblique ejercicios resueltos

We calculate the limit when x tends to 1 from the right, whose result is more infinite:

how to calculate the asynths of a function

As both results are more or less infinite, at x=1 there is a vertical asymptote:

asymptote horizontal

As I said before, the limits do not coincide and there is no limit when x tends to 1, but there is a vertical asymptote.

We will continue with the other number that does not belong to the domain: -1.

We calculate the limit of the function when x tends to -1:

exercises of oblique asynths

We replace the x with -1 and arrive at the indeterminacy of a number between zero:

cómo sacar asintotas

We resolve the indeterminacy of this limit by using the lateral limits.

We calculate the limit when x tends to -1 on the left, whose result is more infinite:

how to find the asynths of a function

We calculate the limit when x tends to -1 on the right, whose result is less infinite:

vertical asynthesis exercises

As before, the limits do not coincide, but as both give a more or less infinite result, then at x=-1 there is a vertival asymptote:

horizontal and vertical asynthesis exercises

Calculation of oblique asymptotes

In this case, since there is a horizontal asymptote, there is no direct oblique asymptote.

Anyway, if we were to calculate it without realizing it, it would be worth 0, so we would be recalculating the horizontal asymptote.

If we represent graphically the function and its asymptotes we can see the function in red and the horizontal and vertical asymptotes in blue. Notice how the function approaches asymptotes but never touches them:

asintota horizontal ejercicios resueltos

We’re going to solve another asymptote calculation exercise.  Find the asymptotes for the next show:

asintotas exercises solved with graphics

Calculation of horizontal asymptotes

We calculate the limit of the function when x tends to infinity:

exercises resolved from vertical and horizontal asynths pdf

We replace the x with infinity and arrive at an indeterminacy:

how to remove the horizontal asynth

We solve the indeterminacy and we obtain as infinite result:

vertical and horizontal asynthesis exercises

Then with infinity, we found no horizontal asymptote.

We calculate the limit of the function when x tends to be less infinite and the result is less infinite:

how to find asintotas

As neither of the two limits has resulted in a finite number, the function has no horizontal asymptotes.

Calculation of vertical asymptotes

To calculate the vertical asymptotes, we calculate the domain of the function, which is:

asymptote

The number that does not belong to the domain is the possible candidate to be a vertical asymptote, which in this case is 3.

We calculate the limit of the function when x tends to 3:

how to calculate asynths

We replace the x with 3 and arrive at an indeterminacy of number between zero:

asintotas horizontal and vertical exercises resolved

We calculated the lateral limits.

The limit when x tends to 3 from the left, whose result is less infinite:

asymptote exercises

The limit when x tends to 3 from the right, whose result is more infinite:

ejemplo de asintotas

The result of the lateral limits does not coincide, but as both give a more or less infinite result, then at x=3 there is a vertical asymptote:

asintota vertical ejercicios resueltos

Calculation of oblique asymptotes

As the function has no horizontal asymptotes, it will have oblique asymptotes, whose equation will be:

horizontal asynthesis exercises

We will then calculate the coefficients m and n.

We begin with the coefficient m that we calculate with the formula:

exercises resolved asintotic

That substituting f(x) for our function remains:

find the asynths of a function

We operate by dividing fractions and then multiplying them into the denominator:

asintotic exercises resolved

We now replace the x with infinity and arrive at indeterminacy:

asintotas horizontales ejemplos

We leave the term of the highest degree in the numerator and in the denominator and arrive at the result of the limit:

asintotic problems

Therefore, in this case, m is equal to 1:

asintotas horizontales como se calculan

Let us now obtain the coefficient n, which we calculate with the formula:

exercises asintotas 1 bachillerato

We replace f(x) and the coefficient m that we have just calculated and that remains:

asintotas horizontales

We operate within the bracket, obtaining a common denominator for subtraction and then grouping terms in the numerator:

asintotas verticales ejercicios

We replace the x with infinity and it remains:

calculate the asynths of a function

We resolve this indeterminacy and arrive at the result:

calculate asynths of a function

The coefficient n equals 3:

equations of asynths

Therefore, after replacing the values of the coefficients m and n in the equation of the line, we are left with the oblique asymptote:

asintotic exercise

If we graphically represent the function, the vertical asymptote and the oblique asymptote, it looks like this:

asintotas verticales y horizontales ejercicios