Asymptotes (vertical, horizontal, removable)
Asymptotes of Rational Functions
Asymptotes of Rational Functions is a topic in Rational Functions in the Texas Essential Knowledge and Skills (Algebra II, §111.40). It is aligned to Standard 2A.6(G), which requires students to analyze the asymptotic behavior of rational functions.
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.