How to Find Vertical & Horizontal Asymptotes? One of the most interesting topics of the study of the analysis of functions of the last years of high school (and first of career) is the representation of functions of a variable. And among the calculations that are considered necessary to collect enough data for the representation is the calculation of the asymptotes of the function.
In this article, very suitable considering the dates we are in (close to the September exams), we are going to see how to do this calculation.
How to Find Vertical & Horizontal Asymptotes?
Definition & Types
We can define the concept of asymptote as follows:
Given a function y = f (x)whose graph is the curve C, the line R is said to be an asymptote of f (x) whether the curve C approaches R indefinitely without coinciding with its own R.
Taking into account that an asymptote is, in particular, a line, we are going to distinguish three types of asymptotes:
- Horizontal Asymptotes
- Vertical Asymptotes
- Oblique Asymptotes
The horizontal asymptotes of a function are horizontal lines of the form y = a. A function can have at most two horizontal asymptotes: one on the left and one on the right .
They are calculated as follows:
Therefore we can find the following cases:
- Functions that do not have horizontal asymptotes.
For example, it f(x) = X3 satisfies that the two limits stated above result in and respectively. We see its graph:
Functions that have a horizontal asymptote that is only on one side
For example, In this case , so the line is a horizontal asymptote from both the left and the right. We see its graph next to its asymptote (in blue):
Functions that have a horizontal asymptote that is on both sides
For example , In this case , so the line is a horizontal asymptote from both the left and the right. We see its graph next to its asymptote (in blue):
Functions that have two different horizontal asymptotes
For example, it fulfills that , so it is a horizontal asymptote from the left and , therefore, it is a horizontal asymptote from the right. You can see its graph along with its two asymptotes (in blue) in the following image:
The vertical asymptotes of a function are vertical lines of the form x = k. There are no restrictions on the number of vertical asymptotes that a function can have: there are functions that do not have vertical asymptotes, functions that have only one, functions that have two, and even functions that have infinite. They are calculated as follows:
One of the conclusions that can be drawn from this is the following: in the horizontal asymptotes we always pose the same limits and the result is the one that tells us whether they exist or not; however, in the verticals, we have to provide the values of K for which to calculate the limits. Obviously we must provide points for which the existence of a vertical asymptote is feasible (it is not too advisable to try random values).
The candidate values for the existence of a vertical asymptote are the following:
Values that override any denominator of the function
For example, for we have a candidate for vertical asymptote at the point x = 1.
Domain interval ends that do not belong to the domain itself
For example, the domain of is the interval . Therefore, it x = 0 is a vertical asymptote candidate for this function.
Consequently, the first thing we must do when we have to calculate the asymptotes of a function is to calculate its domain (fundamental for any calculation related to the graph of a function) and set all denominators that appear in it to zero to collect all the candidates.
Let’s see some interesting cases that may occur:
Functions that do not have vertical asymptotes
For example, if f (x) = sen (x)does not have vertical asymptotes (its domain is R and there are no denominators):
Functions that have a vertical asymptote on both sides
For example, you have a vertical asymptote candidate in x = -1(override the denominator). If we calculate the limits that we have previously discussed, we obtain the following results:
Therefore the line x = 1 is a vertical asymptote f (x) on both sides. We see it in its graph (the asymptote is the blue line):
Functions that have a vertical asymptote on one side only
For example, you have a vertical asymptote candidate in x = 0 (override the two denominators that the function has). We calculate the limits:
Therefore the line x = 0 is a vertical asymptote f (x) only for the right side of the line (for the side that the corresponding limit gives ). We see the graph of the function to the left and to the right of x = 0:
Functions that have infinite vertical asymptotes
We have discussed before that a function can have any number of vertical asymptotes. The possibly most curious case is that of a function that has infinite asymptotes of this type. The best-known example is that of the function f (x) = tan (x). The reason is as follows:
The oblique asymptotes of a function are oblique lines, that is, lines of the form y = mx + n. A function can have a maximum of two different oblique asymptotes (one on the left of its graph and the other on the right of it). The calculation of the same is done as follows:
We can then find the following cases:
Functions that do not have oblique asymptotes
For example, the function does not have oblique asymptotes since when calculating m both left and right we obtain . Its graph is the parabola that we tend to find most often:
Functions that have an oblique asymptote on both sides
For example, the function has a single oblique asymptote, which is also on both sides. Let’s see what exactly is said asymptote:
Functions that have an oblique asymptote on one side only
Curious case, difficult to find on the other hand. An example (taken from the entry on asymptotes on English Wikipedia) could be the function . Its graph is:
Functions that have two different oblique asymptotes
Although it is not easy to find a function of this type, here I bring you one. Specifically it is the function . This function has two oblique asymptotes, namely the line y = x and the line y = -x. We see them in the following graph in blue next to the graph of the function itself:
Two Big Lies About Asymptotes?
As we have mentioned before, the calculation of the asymptotes of a real function of real variable is part of the high school curriculum. In it, as a general rule (actually from personal experience and comments from my students over the years), we can find two great lies about the asymptotes of a function. Let’s see them and give them an explanation more in line with reality:
A function cannot cut an asymptote of its own
First, lie about asymptotes: A function can cut one of its asymptotes. A clear example of this is function . This function has a horizontal asymptote, y = 0 on both sides. We see it in the following graph.
We see in the image that the function cuts its asymptote infinitely both on one side and on the other.
A function cannot have horizontal and oblique asymptotes at the same time
Second lie about asymptotes: a function can have both horizontal and oblique asymptotes at the same time.
Generally, in high school the following is said:
As you can see there are functions that present the two types of asymptotes. What is true is the following:
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I hope this article on the calculation of the asymptotes of a function is useful for your mathematical tasks related to the study of functions. And you know, any questions will be answered as soon as possible in the comments.