Table of Contents
Types of Asymptotes for Rational Functions
In this chapter we focus on asymptotes of rational functions. A rational function has the form
$$
f(x) = \frac{p(x)}{q(x)},
$$
where $p(x)$ and $q(x)$ are polynomials and $q(x) \neq 0$.
We will look at three main types of asymptotes:
- Vertical asymptotes
- Horizontal asymptotes
- Oblique (slant) asymptotes
Throughout, keep in mind that asymptotes describe the long‑term or edge behavior of the graph, not what happens at every point.
Vertical Asymptotes
A vertical asymptote is a vertical line $x = a$ that the graph approaches but does not cross near $x = a$ (though sometimes it can cross elsewhere). For rational functions, vertical asymptotes come from where the denominator is zero and the function “blows up” to $\infty$ or $-\infty$.
For a rational function
$$
f(x) = \frac{p(x)}{q(x)},
$$
a typical vertical asymptote occurs at $x = a$ if:
- $q(a) = 0$ (the denominator is zero), and
- $p(a) \neq 0$ (the numerator is not zero), and
- the factor $(x - a)$ does not completely cancel between numerator and denominator.
In practice:
- Factor $p(x)$ and $q(x)$.
- Cancel any common factors.
- The remaining zeros of the denominator give the vertical asymptotes.
A point where the denominator is zero but the factor cancels with the numerator is not a vertical asymptote; it is a hole in the graph (a removable discontinuity), treated in more detail in the parent chapter on rational functions.
Near a vertical asymptote $x = a$, the function values grow very large in magnitude:
$$
\lim_{x \to a^-} f(x) = \pm\infty \quad\text{or}\quad \lim_{x \to a^+} f(x) = \pm\infty.
$$
You do not need to compute these limits exactly in most algebra courses; it is enough to know that the function goes to infinity or negative infinity and therefore has a vertical asymptote.
Horizontal Asymptotes
A horizontal asymptote is a horizontal line $y = L$ that the graph approaches as $x$ becomes very large in the positive or negative direction:
$$
\lim_{x \to \infty} f(x) = L \quad\text{or}\quad \lim_{x \to -\infty} f(x) = L.
$$
For rational functions, horizontal asymptotes are determined by comparing the degrees of the numerator and denominator polynomials.
Let:
- $\deg(p) = n$ (degree of the numerator),
- $\deg(q) = m$ (degree of the denominator).
Denote the leading terms as:
$$
p(x) \approx a_n x^n, \quad q(x) \approx b_m x^m,
$$
where $a_n$ and $b_m$ are the leading coefficients.
Then:
- Case 1: $n < m$ (degree of numerator < degree of denominator)
The denominator grows faster, so
$$
\lim_{x \to \pm\infty} f(x) = 0.
$$
Horizontal asymptote: $y = 0$ (the $x$‑axis). - Case 2: $n = m$ (same degree)
The leading terms dominate; the ratio approaches the ratio of leading coefficients:
$$
\lim_{x \to \pm\infty} f(x) = \frac{a_n}{b_m}.
$$
Horizontal asymptote: $y = \dfrac{a_n}{b_m}$. - Case 3: $n > m$ (degree of numerator > degree of denominator)
There is no horizontal asymptote.
(In some special situations—especially when $n = m + 1$—we will get an oblique asymptote instead, discussed next.)
Important: A graph may cross its horizontal asymptote, especially for moderate values of $x$. The asymptote only describes the behavior as $x \to \pm\infty$.
Oblique (Slant) Asymptotes
An oblique (or slant) asymptote is a non‑horizontal, non‑vertical line that the graph approaches as $x$ becomes very large or very negative. For rational functions in standard algebra, we mostly see lines of the form
$$
y = mx + b
$$
with $m \neq 0$.
For rational functions $f(x) = \dfrac{p(x)}{q(x)}$:
- An oblique asymptote occurs when the degree of the numerator is exactly one more than the degree of the denominator:
$$
\deg(p) = \deg(q) + 1.
$$
To find the oblique asymptote:
- Perform polynomial long division (or synthetic division, when appropriate) of $p(x)$ by $q(x)$:
$$
\frac{p(x)}{q(x)} = \text{quotient}(x) + \frac{\text{remainder}(x)}{q(x)}.
$$ - The quotient will be a linear polynomial, say $mx + b$.
- The remainder term $\dfrac{\text{remainder}(x)}{q(x)}$ goes to $0$ as $x \to \pm\infty$ (because the remainder has lower degree than $q(x)$).
- Therefore the line $y = mx + b$ is the oblique asymptote.
A rational function can have at most one oblique asymptote in each direction (toward $+\infty$ and $-\infty$), and if it exists it replaces the horizontal asymptote: you cannot have both a non‑zero‑slope oblique and a horizontal asymptote.
Summary of Degree Rules for Asymptotes
For $f(x) = \dfrac{p(x)}{q(x)}$ with degrees $\deg(p) = n$, $\deg(q) = m$:
- Vertical asymptotes:
Roots of $q(x)$ that do not cancel with $p(x)$ (after factoring and simplifying). - Horizontal asymptotes:
- If $n < m$: $y = 0$.
- If $n = m$: $y = \dfrac{\text{leading coefficient of } p}{\text{leading coefficient of } q}$.
- If $n > m$: none.
- Oblique (slant) asymptotes:
- If $n = m + 1$:
Use polynomial division; the quotient line $y = mx + b$ is the asymptote. - If $n \ge m + 2$:
In algebra courses you typically say there is no horizontal or slant asymptote; the end behavior follows a higher‑degree polynomial shape instead of a line.
These rules give you a quick checklist for sketching the overall shape of a rational function and understanding where the graph will get very large and how it behaves far to the left and right.