## Vertical and horizontal asymptote

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

or

as approaches from either the left or the right. A linear rational function (1) has the vertical asymptote  (assuming ).

In general, when and in (2), the line  becomes a vertical asymptote for (2). If and , the point can be considered as a hole.''

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

as (or ).

A linear rational function (1) has the horizontal asymptote .

In general,

1. if in (2), has the horizontal asymptote .

2. If , has the horizontal asymptote  with the respective leading coefficients  and  of  and .

3. If , has no horizontal asymptote.

Oblique asymptote. A line  is called a oblique asymptote if can be expressed in the form

with as (or ).

Department of Mathematics