Asymptotes: Functions, Types and Examples

An asymptote is a line that a graph of a function approaches but never touches or crosses. Learn in detail and understand the types of asymptotes along with the practice questions for a better grasp of the concept here.

In mathematics, asymptotes are an essential concept, particularly when studying the behavior of curves and functions. Asymptotes help us understand how a function behaves as the input values approach infinity, zero, or other critical points. This blog will explore the concept of asymptotes, methods for determining asymptotes of a function, and their role in analyzing rational functions. We’ll also provide examples to solve asymptotes for different types of functions.

1.0What are Asymptotes?

An asymptote is a line that a graph of a function approaches but never touches or crosses. It describes how the function behaves as the input values approach some critical point or infinity. Asymptotes can be of three types:

1. Vertical Asymptotes: The graph of a function approaches a vertical line but never crosses it.
2. Horizontal Asymptotes: The function approaches a horizontal line as x approaches infinity or negative infinity.
3. Oblique (Slant) Asymptotes: The function approaches a slant line when The numerator's degree is greater than that of the denominator in rational functions.

2.0Asymptotes of a Function

The asymptotes of a function depend on the behavior of the function as x approaches specific values, such as infinity or zero. Understanding these asymptotes helps in graphing functions and analyzing their long-term behavior. The following sections explain how to determine each type of asymptote.

3.0Determining Vertical Asymptotes

Vertical asymptotes arise when the denominator of a function tends to zero while the numerator does not. In rational functions, a vertical asymptote exists at values of x that make the denominator zero, provided these values do not cancel with any factors in the numerator.

Steps to Determine Vertical Asymptotes:

1. Set the denominator of the given rational function equal to zero.
2. Solve for x.
3. If the value of x does not cancel with a factor in the numerator, it represents a vertical asymptote.

Example:

For the function $f(x)=\frac{2x}{x^2-4}$ :

• Factor the denominator: $x^2-4=(x-2)(x+2)$ .
• Set the denominator to zero: x – 2 = 0 and x + 2 = 0, so x = 2 and x = –2.
• As these values do not cancel with any factors in the numerator, vertical asymptotes are present at x = 2 and x = –2.

4.0Determining Horizontal Asymptotes

Horizontal asymptotes indicate how the function behaves as x tends toward infinity or negative infinity. The rules for finding horizontal asymptotes depend on the degrees of the numerator and denominator in rational functions.

Rules for Horizontal Asymptotes:

1. If the degree of the numerator is smaller than the degree of the denominator, the horizontal asymptote is y = 0.
2. If the degree of the numerator matches the degree of the denominator, the horizontal asymptote is $y=\frac{\text{ leading coefficient of the numerator }}{\text{ leading coefficient of the denominator }}$ .
3. If the degree of the numerator is greater/more than the degree of the denominator, there is no horizontal asymptote (but there may be an oblique asymptote).

Example:

For the function $f(x)=\frac{3x^2+5x}{2x^2+4}$ :

• The degrees of the numerator and denominator are both 2.
• The horizontal asymptote is $y=\frac{3}{2}$, which is the ratio of the leading coefficients.

5.0Determining Oblique (Slant) Asymptotes

An oblique asymptote exists when the degree of the numerator is exactly one higher than the degree of the denominator. To determine the oblique asymptote, divide the numerator by the denominator using polynomial division.

Steps to Determine Oblique Asymptotes:

1. Divide the numerator by the denominator.

2. The quotient (ignoring the remainder) represents the equation of the oblique asymptote.

Example:

For the function $f(x)=\frac{x^2+2x+3}{x+1}$:

• Perform polynomial division: $x^2+2x+3$ divided by x + 1.
• The quotient is x + 1, so the oblique asymptote is y = x + 1.

6.0Rational Functions and Asymptotes

Rational functions are functions of the form $f(x)=\frac{P(x)}{Q(x)}$ , where P(x) and Q(x) are polynomials. Asymptotes play a crucial role in analyzing the behavior of rational functions.

Key Points to Remember:

• Vertical asymptotes occur at values of x that make the denominator zero.
• Horizontal asymptotes describe the behavior of the function as x approaches infinity.
• Oblique asymptotes arise when the degree of the numerator is one greater than the degree of the denominator.

By determining these asymptotes, you can sketch the graph of a rational function and predict its behavior for large values of x and near points where the function is undefined.

7.0Solving Asymptotes

To solve asymptotes, follow these steps:

1. Vertical Asymptotes: Set the denominator equal to zero and solve for x.
2. Horizontal Asymptotes: Compare the degrees of the numerator and denominator and apply the relevant rule.
3. Oblique Asymptotes: Use polynomial division when the degree of the numerator is one higher than that of the denominator.

Example Problem: Find the vertical, horizontal, and oblique asymptotes of the function

$f(x)=\frac{3x^2-x+2}{x-1}$ .

Solution:

• Vertical Asymptote: Set x – 1 = 0, so x = 1.
• Horizontal Asymptote: Since the degree of the numerator is higher than that of the denominator, there is no horizontal asymptote.
• Oblique Asymptote: Perform polynomial division of 3 x^2-x+2 by x – 1. The quotient is 3x + 2, so the oblique asymptote is y = 3x + 2.

8.0Solved Example of Asymptote

Example 1: Find the vertical and horizontal asymptotes of the function: $f(x)=\frac{2x^2+3x-5}{x^2-4}$

Solution:

1. Vertical Asymptotes:

• Set the denominator equal to zero: 

$x^2-4=0$

• Solving for x:

(x-2)(x+2)=0  

So, x = 2 and x = –2 are the vertical asymptotes.

2. Horizontal Asymptote:

• Compare the degrees of the numerator and denominator.
• Both the numerator and denominator have a degree of 2.
• Therefore, the horizontal asymptote is given by the ratio of the leading coefficients:

$y=\frac{2}{1}=2$

Answer:  

Vertical asymptotes: x = 2, x = –2   

Horizontal asymptote: y = 2

Example 2: Determine the oblique asymptote of the function:

$f(x)=\frac{3x^2+5x+1}{x+2}$

Solution:

Oblique Asymptote: 

Since the degree of the numerator (2) is exactly one greater than the degree of the denominator (1), an oblique asymptote exists.

Perform polynomial division:

$3x^2+5x+1÷(x+2)$

The quotient is 3x – 1.

Answer:  

The oblique asymptote is y = 3x – 1.

Example 3: Find all asymptotes of the function:

$f(x)=\frac{x^3-3x^2+2}{x^2-1}$

Solution:

1. Vertical Asymptotes:

Set the denominator equal to zero:

$x^2-1=0$

(x-1)(x+1)=0 

So, x = 1 and x = –1 are the vertical asymptotes.

2. Horizontal Asymptote:

The degree of the numerator (3) is greater than the degree of the denominator (2).

There is no horizontal asymptote since the degree of the numerator is higher.

3. Oblique Asymptote:

Perform polynomial division: 

$x^3-3x^2+2÷\left(x^2-1\right)$

The quotient is x – 3.

so y = x – 3 is oblique asymptotes.

Answer:  

Vertical asymptotes: x = 1, x = –1   

Horizontal asymptote: None  

Oblique asymptote: y = x – 3

9.0Practice Questions on Asymptote

1. Determine the vertical and horizontal asymptotes of the function: $f(x)=\frac{3x+2}{x-5}$
2. Find the vertical and oblique asymptotes of the function:  $f(x)=\frac{2x^2+x-1}{x-1}$
3. Identify all asymptotes of the function:  $f(x)=\frac{x^2-9}{x^2-4}$
4. Find the horizontal asymptote of the function:  $f(x)=\frac{4x^3-2x}{2x^3+5x}$
5. For the function below, determine if there are any asymptotes:  $f(x)=\frac{5x+3}{x^2-4x+3}$