The asymptotes of a function can be calculated by investigating the behavior of the graph of the function. However, it is also possible to determine whether the function has asymptotes or not without using the graph of the function. The method for calculating asymptotes varies depending on whether the asymptote is vertical, horizontal, or oblique.
In this article, we will see learn to calculate the asymptotes of a function with examples.
How to find the vertical asymptotes of a function?
The vertical asymptotes of a function can be found by examining the factors of the denominator that are not common with the factors of the numerator. The vertical asymptotes occur at the zeros of these factors.
Given a rational function, we can identify the vertical asymptotes by following these steps:
Step 1: Factor the numerator and denominator.
Step 2: Observe any restrictions on the domain of the function.
Step 3: Simplify the expression by canceling common factors in the numerator and denominator.
Step 4: Find any value that makes the denominator zero in the simplified version. This is where the vertical asymptotes occur.
EXAMPLE 1
Find the vertical asymptotes of the rational function $latex f(x)=\frac{{{x}^2}+2x-3}{{{x}^2}-5x-6}$.
Solution: We start by factoring the numerator and the denominator:
$latex f(x)=\frac{(x+3)(x-1)}{(x-6)(x+1)}$
This function can no longer be simplified. Then, x cannot be either 6 or -1 since we would be dividing by zero. Let’s look at the graph of this rational function:
We can see that the graph avoids vertical lines $latex x=6$ and $latex x=-1$. This occurs because x cannot be equal to 6 or -1. Therefore, we draw the vertical asymptotes as dashed lines:
EXAMPLE 2
Find the vertical asymptotes of the function $latex g(x)=\frac{x+2}{{{x}^2}+2x-8}$.
Solution: The numerator is already factored, so we factor to the denominator:
$latex f(x)=\frac{x+2}{(x+4)(x-2)}$
We cannot simplify this function and we know that we cannot have zero in the denominator, therefore, x cannot be equal to $latex x=-4$ or $latex x=2$. This tells us that the vertical asymptotes of the function are located at $latex x=-4$ and $latex x=2$:
Try solving the following practice problems
How to find the horizontal asymptotes of a function?
The method for identifying horizontal asymptotes changes based on how the degrees of the polynomial compare in the numerator and denominator of the function. To find the horizontal asymptotes, we have to remember the following:
- If the degree of the polynomial in the numerator is equal to the degree of the polynomial in the denominator, we divide the coefficients of the terms with the largest degree to obtain the horizontal asymptotes.
- If the degree of the numerator is less than the degree of the denominator, the horizontal asymptotes will be $latex y=0$.
- If the degree of the numerator is greater than the degree of the denominator, then there are no horizontal asymptotes.
Let’s see some examples:
EXAMPLE 1
Find the horizontal asymptotes of the function $latex g(x)=\frac{x+2}{2x}$.
Solution: Since the largest degree in both the numerator and denominator is 1, then we consider the coefficient of x.
In this case, the horizontal asymptote is located at $latex y=\frac{1}{2}$:
EXAMPLE 2
Find the horizontal asymptotes of the function $latex g(x)=\frac{x}{{{x}^2}+2}$.
Solution: Here, we can see that the degree of the numerator is less than the degree of the denominator, therefore, the horizontal asymptote is located at $latex y=0$:
EXAMPLE 3
Find the horizontal asymptotes of the function $latex f(x)=\frac{{{x}^2}+2}{x+1}$.
Solution: In this case, the degree of the numerator is greater than the degree of the denominator, so there is no horizontal asymptote:
Try solving the following practice problems
How to find the oblique asymptotes of a function?
To find the oblique or slanted asymptote of a function, we have to compare the degree of the numerator and the degree of the denominator.
If the degree of the numerator is exactly one more than the degree of the denominator, then the graph of the rational function will be roughly a sloping line with some complicated parts in the middle. The asymptote of this type of function is called an oblique or slanted asymptote.
We can obtain the equation of this asymptote by performing long division of polynomials. The equation of the asymptote is the integer part of the result of the division.
EXAMPLE
Find the oblique asymptote of the function $latex f(x)=\frac{-3{{x}^2}+2}{x-1}$.
Solution: We start by performing the long division of this rational expression:
At the top, we have the quotient, the linear expression $latex -3x-3$. At the bottom, we have the remainder. This means that, through division, we convert the function into a mixed expression:
$latex f(x)=-3x-3+\frac{-1}{x-1}$
This is the same function, we just rearrange it. When graphing the function along with the line $latex y=-3x-3$, we can see that this line is the oblique asymptote of the function:
Try solving the following practice problem
See also
Interested in learning more about functions? Take a look at these pages:
Jefferson Huera Guzman
Jefferson is the lead author and administrator of Neurochispas.com. The interactive Mathematics and Physics content that I have created has helped many students.
