Precalculus Review

Calculus I > Precalculus Review

Vertical and Horizontal Asymptote

Linear rational functions. A function

is called a linear rational function if

$\displaystyle f(x) = \frac{a x + b}{c x + d}$

Rational functions. A function is called a rational function if

$\displaystyle f(x) = \frac{g(x)}{h(x)}$

(1)

with polynomial functions

of degree

and

of degree

.

Vertical asymptote. A line is called a vertical asymptote for if

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

as

approaches

from either the left or the right. A linear rational function has the vertical asymptote

(assuming $ad \neq bc$ ).

In general, when $g(a) \neq 0$ and in (Equation 1), the line becomes a vertical asymptote for (Equation 1). If and , the point can be considered as a “hole.”

Horizontal asymptote. A line is called a horizontal asymptote for if

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

If , a rational function has the horizontal asymptote .
If , has the horizontal asymptote with the respective leading coefficients and of and .
If , has no horizontal asymptote.

In particular, a linear rational function has the horizontal asymptote .

Oblique asymptote. A line is called a oblique asymptote if 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 $x \to -\infty$ ).

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