Vertical and horizontal asymptote

Vertical asymptote. A line $ x = a$ is called a vertical asymptote for $f$ if

$\displaystyle f(x) \to \infty$    or $\displaystyle \quad
f(x) \to -\infty$

as $x$ approaches $a$ from either the left or the right. A linear rational function (1) has the vertical asymptote  $ x = -(d/c)$ (assuming $ ad \neq bc$).

In general, when $ g(a) \neq 0$ and $ h(a) = 0$ in (2), the line $ x = a$ becomes a vertical asymptote for (2). If $ g(a) = 0$ and $ h(a) = 0$, the point $ x = a$ can be considered as a ``hole.''

Horizontal asymptote. A line $ y = c$ is called a horizontal asymptote for $f$ if

$\displaystyle f(x) \to c$    as $ x \to \infty$ (or $ x \to -\infty$).

A linear rational function (1) has the horizontal asymptote $ y = (a/c)$.

In general,

  1. if $ n < k$ in (2), $f$ has the horizontal asymptote $ y = 0$.

  2. If $ n = k$, $f$ has the horizontal asymptote  $ y = (a_n/b_k)$ with the respective leading coefficients $a_n$ and $ b_k$ of $ g(x)$ and $ h(x)$.

  3. If $ n > k$, $f$ has no horizontal asymptote.

Oblique asymptote. A line  $ y = a x + b$ is called a oblique asymptote if $f$ can be expressed in the form

$\displaystyle f(x) = (a x + b) + \frac{r(x)}{h(x)} ,
$

with $ r(x)/h(x) \to 0$ as $ x \to \infty$ (or $ x \to -\infty$).

Department of Mathematics
Last modified: 2005-09-29