Finding Vertical Asymptotes of Rational Functions
![](/math/calculus/images/finding_vertical_asymptotes_of_rational_functions_img1.png)
In this example, there is a vertical asymptote at x = 3 and a horizontal asymptote at y = 1. The curves approach these asymptotes but never cross them.
To find the vertical asymptote(s) of a rational function, simply set the denominator equal to 0 and solve for x.
Examples: Find the vertical asymptote(s)
![](/math/calculus/images/finding_vertical_asymptotes_of_rational_functions_img2.png)
We mus set the denominator equal to 0 and solve:
x + 5 = 0
x = -5
There is a vertical asymptote at x = -5
x + 5 = 0
x = -5
There is a vertical asymptote at x = -5
![](/math/calculus/images/finding_vertical_asymptotes_of_rational_functions_img3.png)
We mus set the denominator equal to 0 and solve:
![](/math/calculus/images/finding_vertical_asymptotes_of_rational_functions_img4.png)
This quadratic can most easily be solved by factoring the trinomial and setting the factors equal to 0.
![](/math/calculus/images/finding_vertical_asymptotes_of_rational_functions_img5.png)
![](/math/calculus/images/finding_vertical_asymptotes_of_rational_functions_img6.png)
![](/math/calculus/images/finding_vertical_asymptotes_of_rational_functions_img7.png)
There are vertical asymptotes at
.
![](/math/calculus/images/finding_vertical_asymptotes_of_rational_functions_img4.png)
This quadratic can most easily be solved by factoring the trinomial and setting the factors equal to 0.
![](/math/calculus/images/finding_vertical_asymptotes_of_rational_functions_img5.png)
![](/math/calculus/images/finding_vertical_asymptotes_of_rational_functions_img6.png)
![](/math/calculus/images/finding_vertical_asymptotes_of_rational_functions_img7.png)
There are vertical asymptotes at
![](/math/calculus/images/finding_vertical_asymptotes_of_rational_functions_img8.png)
![](/math/calculus/images/finding_vertical_asymptotes_of_rational_functions_img9.png)
We mus set the denominator equal to 0 and solve:
![](/math/calculus/images/finding_vertical_asymptotes_of_rational_functions_img10.png)
This quadratic can most easily be solved by factoring out the x and setting the factors equal to 0.
x(x - 5) = 0
![](/math/calculus/images/finding_vertical_asymptotes_of_rational_functions_img11.png)
![](/math/calculus/images/finding_vertical_asymptotes_of_rational_functions_img12.png)
There are vertical asymptotes at
.
![](/math/calculus/images/finding_vertical_asymptotes_of_rational_functions_img10.png)
This quadratic can most easily be solved by factoring out the x and setting the factors equal to 0.
x(x - 5) = 0
![](/math/calculus/images/finding_vertical_asymptotes_of_rational_functions_img11.png)
![](/math/calculus/images/finding_vertical_asymptotes_of_rational_functions_img12.png)
There are vertical asymptotes at
![](/math/calculus/images/finding_vertical_asymptotes_of_rational_functions_img13.png)
Practice: Find the vertical asymptote(s) for each rational function:
![](/math/calculus/images/finding_vertical_asymptotes_of_rational_functions_img14.png)
![](/math/calculus/images/finding_vertical_asymptotes_of_rational_functions_img15.png)
![](/math/calculus/images/finding_vertical_asymptotes_of_rational_functions_img16.png)
![](/math/calculus/images/finding_vertical_asymptotes_of_rational_functions_img17.png)
![](/math/calculus/images/finding_vertical_asymptotes_of_rational_functions_img18.png)
Answers: 1) x = -4 2) x = 6 and x = -1 3) x = 0 4) x = 0 and x = 2 5) x = -3 and x = -4
Related Links: Math Fractions Factors |