A vertical asymptote usually occurs where a simplified rational function's denominator equals zero. A horizontal asymptote describes the function's end behaviour as becomes very large positively or negatively.
Formally, is vertical when at least one one-sided limit is infinite, while is horizontal when or . The two ends may have different horizontal asymptotes.
For a rational function , simplify it first. Then solve ; any resulting value is a vertical asymptote if grows without bound as approaches .
To find a horizontal asymptote, compare the degrees and leading coefficients of and . These rules follow by dividing the numerator and denominator by the highest relevant power of : lower-order terms approach zero, leaving only the dominant terms.
| Comparison | Horizontal asymptote |
|---|---|
| Degree of numerator is less than degree of denominator | |
| Degrees are equal | |
| Degree of numerator is greater than degree of denominator | No horizontal asymptote; an oblique asymptote may exist |
For example, consider
The denominator is zero when , so the vertical asymptote is . The degrees are equal, and the ratio of leading coefficients is , so the horizontal asymptote is . Equivalently, polynomial division gives , and the fraction approaches as (|x|) grows.
A common misconception is that every denominator zero gives a vertical asymptote. For , cancellation produces a hole at , not a vertical asymptote. Always simplify before classifying the discontinuity. A graph can also cross a horizontal asymptote because it describes end behaviour only.
This is IB Mathematics AA SL, mainly sub-topic SL 2.8: Reciprocal and simple rational functions, with links to SL 2.4: Key features of graphs. On Paper 2, use your GDC to identify asymptotes, but support the result algebraically and label them explicitly as or .
