Rational Functions
Asymptotes & Holes
Finding vertical, horizontal, and removable discontinuities of a rational function.
Two Different Ways a Function Can Break
f(x) = 1/(x − 3) and g(x) = (x − 3)(x + 1)/(x − 3) are both undefined at x = 3. Before reading on, think about whether their graphs should look the same near x = 3 — and consider what's different about g's formula that might change what happens there.
Definition — Vertical Asymptote and Removable Discontinuity (Hole)
f(x) = 1/(x − 3) has nothing to cancel — as x approaches 3, the denominator shrinks toward 0 while the numerator stays 1, forcing the fraction's size to explode toward infinity: a vertical asymptote. But g(x) = (x − 3)(x + 1)/(x − 3) simplifies, for every x other than 3, to just (x + 1) — the (x − 3) factors cancel algebraically, so nearby values behave exactly like the simple line y = x + 1. The function is still technically undefined at x = 3 (since the original expression involves 0/0 there), but it doesn't blow up — it's a single missing point, a hole, not an asymptote.
Worked Example — Distinguishing an Asymptote from a Hole
A rational function's horizontal asymptote — its behavior as x → ±∞ — follows the same dominant-term reasoning from polynomial end behavior, applied to a ratio: compare the degree of the numerator to the degree of the denominator. If the denominator's degree is larger, the whole fraction shrinks toward 0. If the degrees match, the fraction approaches the ratio of the leading coefficients. If the numerator's degree is larger, there's no horizontal asymptote at all — the function grows without bound.
Worked Example — Finding a Horizontal Asymptote
Function Explorer
Transform: g(x) = a·f(b(x − h)) + k
Composition
Analysis (of the transformed function, in view)
- y-intercept
- (0, 0)
- x-intercepts
- (-9.42, 0), (-6.28, 0), (-3.14, 0), (0, 0), (3.14, 0), (6.28, 0), (9.42, 0)
- Extrema
- local min at (-7.85, -1); local max at (-4.71, 1); local min at (-1.57, -1); local max at (1.57, 1); local min at (4.71, -1); local max at (7.85, 1)
- Inflection points
- (-9.42, 0), (-6.28, 0), (-3.14, 0), (0, 0), (3.14, 0), (6.28, 0), (9.42, 0)
- Vertical asymptotes
- none found in view
- Horizontal asymptotes
- none found
- Domain
- all real numbers in view
- Range (estimated)
- approximately [-1, 1]
Tip
Common Mistakes
Treating every x-value that zeros the denominator as a vertical asymptote, without first checking whether the same factor also appears in the numerator.
Factor both the numerator and denominator first — a factor that cancels between them creates a hole, not an asymptote, even though the original, unfactored expression is technically undefined there too.
Comparing numerator and denominator degrees incorrectly when finding a horizontal asymptote, such as assuming there's always a horizontal asymptote regardless of degree comparison.
A horizontal asymptote exists only when the denominator's degree is greater than or equal to the numerator's — if the numerator's degree is larger, the function has no horizontal asymptote at all.
Key Takeaways
- A vertical asymptote occurs where the denominator is 0 and the corresponding factor doesn't cancel; a hole occurs where it does.
- Factoring completely first reveals whether a given x-value is truly an asymptote or actually a removable hole.
- A horizontal asymptote's existence and value depend on comparing the numerator's and denominator's degrees.
Summary
Asymptotes and holes reveal exactly how a rational function's algebraic structure shapes its graph. The final lesson in this unit solves equations built from rational expressions.
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