Horizontal Asymptote Rules: The Root and Explanation of the Term ‘’Asymptote’’


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Horizontal Asymptote Rules: The Root and Explanation of the Term ‘’Asymptote’’

The term horizontal asymptote rules may sound quite familiar to math specialists and yet it may seem a strange and mysterious combination of words for people with no idea about this sphere and asymptotes in general. If you are among such people belonging to the second group then you should learn and remember that asymptote is a line that the curve approaches and yet does not cross on the graph. This rule works regardless of how far we go into infinity because the line will never actually reach the curve. However, there are other definitions of this term as well like asymptote is a line that is tangent to a curve at infinity. The word asymptote comes from the Greek word á¼Â€σύμπτωτος meaning ’not falling together’. The first person who introduced this term was Apollonius of Perga who is the author of the work devoted to conic sections but in those days that term held a bit different meaning than the one used today. That is to say, the word asymptote was used to define any line not intersecting the specifically given curve. Asymptotes carry essential information about the behavior of curves in the large which means that determining the asymptotes of a function is a really important step in sketching the graph. Note, that the study of asymptotes of functions forms a part of asymptotic analysis.

Don’t you think that before passing on to the main topic, which is discussing horizontal asymptote rules we should have an idea about the term horizontal asymptote for the first place? Sure you do! In the meantime when the vertical asymptotes are sacred ground horizontal asymptotes are considered to be just useful suggestions. If in the case of a vertical asymptote you will not be able to touch them, you can touch and sometimes even cross horizontal asymptotes. It is an interesting fact that whereas vertical asymptotes are used to show very specific behavior on the graph the horizontal asymptotes point out only general behavior far off to the sides of the graph.

Horizontal Asymptote Rules: Types Of Asymptotes

There are three main types of asymptotes - horizontal, vertical, and oblique. Keep in mind that the horizontal one can be considered as an asymptote only in case it is to the far left or the far right of the graph. Wondering what the far left and the far right mean? These terms are usually defined as anything that is past either the vertical asymptotes or x-intercepts. Horizontal asymptotes can never be asymptotic in the middle. However, it is more than ok to cross a horizontal asymptote located in the middle.

Now it is time to explain how we can determine the horizontal asymptote’s location just by looking at the degrees of the numerator (n) and denominator (m). Here are a few examples showing you how to point out the horizontal asymptote easily and without any problems. Let’s assume that our nm. Do we have any horizontal asymptote in this case as well? The answer is definitely no!

The second type of asymptote is already mentioned - the vertical one. If you are in search of the equations of the vertical asymptotes then you should know that they can be noticed by finding the roots of q(x). In the case of looking for the vertical asymptotes you are free to completely ignore the numerator. In this case, you should attach importance only to the denominator. In case you prefer having them in a form of factor, it will be easier for you to tell whether the graph is asymptotic in the same direction or it is not by paying attention to the multiplicity that can be either even or odd. If the asymptote is in the same direction it means that the curve will go up and down on vertical asymptote’s both the left and right sides. In case we deal with asymptote in different directions it means that the curve’s one side will go up and down while the other side will go up at the vertical asymptote.

However, please, note that in the equation n=m+1 you will find an oblique asymptote which is often called slant asymptote as well. To find the oblique asymptote’s equation what you really need to do is to perform a long division just by dividing the denominator into nominator. With x getting very large (this was the far left and the far right we have already talked about) the remaining portion is very small that is almost zero.

Horizontal Asymptote Rules: Methods of Finding Horizontal Asymptotes

When we talk about any phenomenon it seems really easy to understand it and yet when it comes to real examples we can stay stuck unable to solve the problem practically. That is the reason why we have decided to bring some examples involving the horizontal asymptote rules to you showing exactly how you can find horizontal asymptotes whenever you are given such a task. Always revise horizontal asymptotic rules so that solving such equations becomes easier and easier for you with each passing day! Let’s try to find the horizontal asymptote in the example given below. 

Here is our last, the third example, which is ready to help you gain more practical knowledge about horizontal asymptote rules. We will compare the degrees of the polynomials once more finding out that the numerator contains the 1st-degree polynomial whereas the denominator contains the 3rd one. As we can see in this example, the polynomial of the nominator is of a lower degree than the one found in the denominator which means that the horizontal asymptote is located at y=0. As you can see there is nothing scary in the topic of horizontal asymptotes. The only thing you need to do is to study the subject thoroughly, learn all the rules perfectly and read more interesting articles on horizontal asymptote rules.

Originally published Feb 18, 2021, updated Feb 24, 2021

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