How To Find Vertical And Horizontal Asymptotes Of A Rational Function

The asymptotes of a part can exist calculated by investigating the behavior of the graph of the role. Still, it is also possible to determine whether the office 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 encounter acquire to calculate the asymptotes of a function with examples.

ALGEBRA

Relevant for…

Learning to discover the three types of asymptotes.

See method

ALGEBRA

Relevant for…

Learning to find the three types of asymptotes.

See method

How to find the vertical asymptotes of a function?

The vertical asymptotes of a office tin can be found by examining the factors of the denominator that are non common with the factors of the numerator. The vertical asymptotes occur at the zeros of these factors.

Given a rational office, we can place the vertical asymptotes by post-obit these steps:

Step ane: Cistron the numerator and denominator.

Step 2: Observe whatever restrictions on the domain of the function.

Step 3: Simplify the expression by canceling common factors in the numerator and denominator.

Step 4: Observe any value that makes the denominator aught in the simplified version. This is where the vertical asymptotes occur.

Case i

Find the vertical asymptotes of the rational function $latex f(x)=\frac{{{x}^2}+2x-iii}{{{10}^2}-5x-half dozen}$.

Solution: We first by factoring the numerator and the denominator:

$latex f(ten)=\frac{(10+iii)(x-1)}{(10-six)(x+1)}$

This office can no longer be simplified. Then,x cannot exist either 6 or -1 since nosotros would exist dividing past zilch. Let'southward await at the graph of this rational function:

We can see that the graph avoids vertical lines $latex ten=6$ and $latex x=-ane$. This occurs becausex cannot be equal to half-dozen or -one. Therefore, we draw the vertical asymptotes equally dashed lines:

vertical asymptotes of a rational function 1

EXAMPLE two

Find the vertical asymptotes of the function $latex g(x)=\frac{10+two}{{{10}^2}+2x-eight}$.

Solution: The numerator is already factored, so we factor to the denominator:

$latex f(x)=\frac{10+2}{(ten+iv)(x-2)}$

Nosotros cannot simplify this function and we know that nosotros cannot accept zero in the denominator, therefore,ten cannot be equal to $latex x=-iv$ or $latex ten=ii$. This tells us that the vertical asymptotes of the part are located at $latex x=-4$ and $latex ten=2$:

vertical asymptotes of a rational function 2

Start now: Explore our additional Mathematics resource

Try solving the following practice problems

Find the vertical asymptote of $latex f(10)=\frac{{{x}^iii}-viii}{{{x}^ii}+5x+vi}$.

Cull an respond

$latex x=iii$, $latex 10=2$

$latex ten=-iii$, $latex x=-ii$

$latex ten=-two$, $latex ten=3$

$latex 10=ane$, $latex x=-2$

How to find the horizontal asymptotes of a role?

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 accept to remember the following:

If the caste 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 caste 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 at that place are no horizontal asymptotes.

Let's see some examples:

EXAMPLE 1

Detect the horizontal asymptotes of the office $latex one thousand(x)=\frac{x+2}{2x}$.

Solution: Since the largest caste in both the numerator and denominator is 1, then we consider the coefficient ofx.

In this case, the horizontal asymptote is located at $latex y=\frac{1}{ii}$:

horizontal asymptote of a rational function 1

Example 2

Find the horizontal asymptotes of the role $latex g(x)=\frac{x}{{{x}^2}+2}$.

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 $latex y=0$:

horizontal asymptote of a rational function 2

EXAMPLE iii

Find the horizontal asymptotes of the function $latex f(ten)=\frac{{{x}^ii}+ii}{x+1}$.

Solution: In this case, the caste of the numerator is greater than the degree of the denominator, so in that location is no horizontal asymptote:

Try solving the following practice bug

Find the horizontal asymptote of $latex f(x)=\frac{x+2}{{{10}^2}+1}$.

Cull an answer

$latex y=2$

$latex y=1$

$latex y=0$

$latex y=-1$

Find the horizontal asymptote of $latex f(x)=\frac{four{{x}^iii}+3x}{five-three{{x}^3}}$.

Choose an answer

$latex y=-\frac{4}{5}$

$latex y=\frac{four}{5}$

$latex y=\frac{4}{three}$

$latex y=-\frac{4}{3}$

How to find the oblique asymptotes of a function?

To observe the oblique or slanted asymptote of a function, we have to compare the caste of the numerator and the degree of the denominator.

If the degree of the numerator is exactly one more than 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 chosen an oblique or slanted asymptote.

We can obtain the equation of this asymptote past performing long division of polynomials. The equation of the asymptote is the integer part of the result of the division.

Case

Find the oblique asymptote of the office $latex f(ten)=\frac{-3{{10}^2}+2}{x-i}$.

Solution: Nosotros offset past performing the long division of this rational expression:

At the top, we have the quotient, the linear expression $latex -3x-iii$. At the bottom, we have the rest. This means that, through division, we convert the function into a mixed expression:

$latex f(x)=-3x-three+\frac{-1}{ten-1}$

This is the aforementioned function, we just rearrange information technology. When graphing the function along with the line $latex y=-3x-three$, we can see that this line is the oblique asymptote of the office: