Asymptotes

Asymptotes

Asymptotes, it appears, believe in the famous line: to infinity and beyond, as they are curves that do not have an end.

While understanding asymptotes, you would have chanced upon a graph that reads  \(f(x)=\frac{1}{x}\)

You might have observed a strange behavior at x=0.

graph of the equation 1/x at x=0

Let us explore that a little more about that strange behavior, while we learn about asymptotes.

Lesson Plan 

What Is Meant by Asymptotes?

An asymptote is a line being approached by a curve but doesn't meet it infinitely.

In other words, you can say that asymptote is a line to which the curve converges. The asymptote never crosses the curve even though they get infinitely close.


What Are the Different Types of Asymptotes?

There are 3 types of asymptotes.

  1. Horizontal 
  2. Vertical
  3. Oblique
Horizontal Vertical Oblique
horizontal asymptote vertical asymptote oblique or angular asymptote

How to Find Vertical and Horizontal Asymptotes in a Function?

A graph can have more than one vertical or horizontal asymptote.

Vertical Asymptote

The straight-line x=a is a vertical asymptote of the graph of the function y=f(x) if at least one of these conditions is true:

\(\lim _{x \rightarrow a+} f(x)=\pm \infty, \quad \lim _{x \rightarrow a-} f(x)=\pm \infty\)

A vertical asymptote occurs in rational functions at the points when the denominator is zero and the numerator is not equal to zero.

We can find the vertical asymptote by equating the denominator of the rational function to zero.

For example, consider the function \(\dfrac{(x-5)}{(x+4)}\)

\(\lim _{x \rightarrow -4+} f(x)=\pm \infty, \quad \lim _{x \rightarrow -4-} f(x)=\pm \infty\)

so, there exists a vertical asymptote at x=-4.

Vertical Asymptote is obtained when we equate the denominator to zero.

 \begin{eqnarray}
x+4&=& 0\\
x &=&-4
\end{eqnarray}

Horizontal Asymptote

The method to find the horizontal asymptote changes based on the degrees of the polynomials in the numerator and denominator of the function.

  1. If both the polynomials have the same degree, divide the coefficients of the largest degree terms. This is your asymptote!
  2. If the degree of the numerator is less than the denominator, then the asymptote is located at y=0.
  3. If the degree of the numerator is greater than the denominator, then there is no horizontal asymptote! 

For example, consider the function \(\dfrac{(x-5)}{(x-4)}\)

In this case, we have the horizontal asymptote at the point y=1 as it falls under case-1.(numerator and denominator are of same degree: linear)


Solved Examples 

Example 1

 

 

Find the asymptotes of the graph:\(f(x)=\frac{x^{2}-3 x}{x-5}\)

Solution

The vertical asymptote is obtained at x=5 as \(\lim _{x \rightarrow 5+} f(x)=\pm \infty, \quad \lim _{x \rightarrow 5-} f(x)=\pm \infty\)

Here there is no horizontal asymptote following the case-3 rule.

as the degree of numerator = 2

and the degree of denominator = 1

d(N)>d(D).

asymptotes of graph f(x)=\frac{x^{2}-3 x}{x-5}

\(\therefore\) There is no horizontal asymptote but it has a vertical asymptote at x=5
Example 2

 

 

Find the vertical asymptote of the function f(x)=\(\frac{x^{2}-3 x+3}{x^{2}-5}\)

Solution

The vertical asymptote of the graph is obtained at the point where the denominator \(x^{2}-5\) is equal to zero.

 \begin{eqnarray}
{x^{2}-5} &=& 0\\
x &=& -\sqrt{5}, \sqrt{5}
\end{eqnarray}

vertical asymptote

 \(\therefore\) Vertical asymptotes occur at \(x = -\sqrt{5}, \sqrt{5}\)
Example 3

 

 

Find the asymptotes of \(f(x)=\frac{1}{x^{2}-3 x+2}\)

Solution

The vertical asymptote occurs at the point  \(x^{2}-3 x+2\) equals to zero.

\begin{eqnarray}
x^{2}-3 x+2 &=& 0\\
(x-1)(x-2) &=& 0\\
\implies \text {assymptote occurs at} \ x&=&1\text (or) x=2
\end{eqnarray}

assymptote of the graph

 \(\therefore\) Vertical asymptotes occur at x = 1,2
Example 4

 

 

Find the horizontal asymptote of \(\frac{\left(x^{2}-4 x+5\right)}{x-6}\),if it exists.

Solution

The vertical asymptote occurs at x=6.

There is no horizontal asymptote as the degree of the numerator is more than that of the denominator.

horizontal asymptote

\(\therefore\) There is no horizontal asymptote for this graph.
Example 5

 

 

Can a graph have no vertical or horizontal asymptotes?

Solution

Yes, for example, \(\frac{\left(x^{2}-5 x+6\right)}{x-2}\) does not have any vertical or horizontal asymptote.

graph with no vertical or horizontal asymptoes

\(\therefore\) Yes,\(\frac{\left(x^{2}-5 x+6\right)}{x-2}\)does have only oblique asymptote.
 
Thinking out of the box
Think Tank
  1. Find a function with no asymptotes at all?
  2. Can a continuous function have vertical asymptotes?

Asymptote Calculator for Any Graph

 
Challenge your math skills
Challenging Questions
  1. Find the asymptotes of \(f(x)=x+\frac{1}{x}\)

Interactive Questions

Here are a few activities for you to practice. Select/Type your answer and click the "Check Answer" button to see the result.
 
 
 

Let's Summarize

This mini-lesson targeted the fascinating concept of asymptotes. The math journey around the asymptotes starts with what a student already knows, and goes on to creatively crafting a fresh concept in the young minds. Done in a way that not only it is relatable and easy to grasp, but also will stay with them forever. Here lies the magic with Cuemath.

About Cuemath

At Cuemath, our team of math experts is dedicated to making learning fun for our favorite readers, the students!

Through an interactive and engaging learning-teaching-learning approach, the teachers explore all angles of a topic.

Be it worksheets, online classes, doubt sessions, or any other form of relation, it’s the logical thinking and smart learning approach that we, at Cuemath, believe in.


Frequently Asked Questions (FAQs) :

1. What are asymptotes used for?

Asymptotes are used to study the behavior of curves in the large, and determining the asymptotes of a function plays a vital role in drawing its graph.

2. Do square root functions have asymptotes?

The square root functions which are non-rational functions do not have asymptotes.

As they are continuous over its domain.

3. Do polynomials have asymptotes? 

A polynomial has no vertical or horizontal asymptotes, or of the form y = mx + b (linear)

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