Asymptotes (vertical, horizontal, removable)
Asymptotes of Rational Functions
Asymptotes of Rational Functions is a topic in Rational Functions in the California Common Core State Standards. It is aligned to Standard F-IF.7d, which requires students to graph rational functions, identifying zeros and asymptotes when suitable factorizations are available.
Vertical asymptotes come from uncancelled denominator zeros and horizontal asymptotes from comparing degrees.
Theory
A rational function's asymptotes come from its structure:
- Vertical: where an uncancelled denominator factor is zero.
- Horizontal: compare degrees — top \(<\) bottom gives \(y=0\); equal gives the ratio of leading coefficients.
- Removable (hole): a factor that cancels.
Horizontal asymptote by degree:
How to find asymptotes
- Factor and cancel common factors (note holes).
- Set the remaining denominator to zero for vertical asymptotes.
- Compare degrees for the horizontal asymptote.
- If top degree is one more, divide for a slant asymptote.
Denominator zero at \(x=2\); degree top \(<\) bottom.
| \(\text{VA}\) | \(:\) | \(x=2\) |
| \(\text{HA}\) | \(:\) | \(y=0\) |
Degrees are equal, so use the ratio of leading coefficients.
| \(\text{HA}\) | \(:\) | \(y=\dfrac{2}{1}=2\) |
The factor \((x-1)\) cancels, leaving a hole.
| \(\dfrac{(x-1)(x+1)}{x-1}\) | \(=\) | \(x+1,\ x\neq1\) |
There is a hole at \(x=1\), not an asymptote.
Degree of top is one more than bottom; divide.
| \(\dfrac{x^2}{x-1}\) | \(=\) | \(x+1+\dfrac{1}{x-1}\) |
| \(\text{slant}\) | \(:\) | \(y=x+1\) |
Common pitfalls
Frequently asked questions
Where are vertical asymptotes?
Where an uncancelled denominator factor equals zero.
How do you find a horizontal asymptote?
Compare the degrees of numerator and denominator.
What is a removable discontinuity?
A hole, created by a factor that cancels.
When is there a slant asymptote?
When the numerator's degree is exactly one more than the denominator's.