﻿ ▷ Asymptotes of a function. Exercises resolved step by step.

# 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:

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

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:

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:

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:

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

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:

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:

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

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

The coefficient n is calculated with the following formula:

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:

## 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:

Calculation of horizontal asymptotes

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

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

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

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

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

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

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

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

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:

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:

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:

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:

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

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

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:

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

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:

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

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:

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:

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

Calculation of horizontal asymptotes

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

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

We solve the indeterminacy and we obtain as infinite result:

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:

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:

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:

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

We calculated the lateral limits.

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

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

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:

Calculation of oblique asymptotes

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

We will then calculate the coefficients m and n.

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

That substituting f(x) for our function remains:

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

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

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

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

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

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

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

We replace the x with infinity and it remains:

We resolve this indeterminacy and arrive at the result:

The coefficient n equals 3:

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

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