http://www.ck12.org Chapter 2. Polynomials and Rational Functions
2.9 Vertical Asymptotes
Here you will learn to recognize when vertical asymptotes occur and what makes them different from removable
discontinuities.
The basic rational functionf(x) =^1 xis a hyperbola with a vertical asymptote atx=0. More complicated rational
functions may have multiple vertical asymptotes. These asymptotes are very important characteristics of the function
just like holes. Both holes and vertical asymptotes occur atxvalues that make the denominator of the function
zero. A driving question is: what makes vertical asymptotes different from holes?
Watch This
Watch the following video, focusing on the parts about vertical asymptotes.
MEDIA
Click image to the left for use the URL below.
URL: http://www.ck12.org/flx/render/embeddedobject/60833http://www.youtube.com/watch?v=wBZxVxiJS9I James Sousa: Determining Horizontal and Vertical Asymptotes
of Rational Functions
Guidance
Vertical asymptotes occur when a factor of the denominator of a rational expression does not cancel with a factor
from the numerator. When you have a factor that does not cancel, instead of making a hole at thatxvalue, there
exists a vertical asymptote. The vertical asymptote is represented by a dotted vertical line. Most calculators will not
identify vertical asymptotes and some will incorrectly draw a steep line as part of a function where the asymptote
actually exists.
Your job is to be able to identify vertical asymptotes from a function and describe each asymptote using the equation
of a vertical line.
Example A
Identify the holes and the equations of the vertical asymptotes for the following rational function.
f(x) =(^2 x−(x^3 +)( 2 x)(+x^1 +)( 1 x)−^2 )
Solution: The factor that cancels represents the removable discontinuity. There is a hole at (-1, 15). The vertical
asymptote occurs atx=−2 because the factorx+2 does not cancel.
Example B
Identify the domain of the following function and then identify the holes and vertical asymptotes.
f(x) =(^3 x−( 34 x)(−^12 −)(xx)(−x 12 )+^4 )Solution: The domain of the function written in interval notation is:(−∞,^23 )∪(^23 , 1 )∪( 1 ,∞)