Asymptote

A line that a curve approaches, as it heads towards infinity:

Asymptote

Types

There are three types: horizontal, vertical and oblique:

Asymptote Types

 

It can be in a negative direction,

the curve can approach from any side (such as from above or below for a horizontal asymptote),

Asymptote Crossing

or may actually cross over (possibly many times), and even move away and back again.

The important point is that:

The distance between the curve and the asymptote tends to zero as they head to infinity (or −infinity)

Horizontal Asymptotes

Horizontal Asymptote  

It is a Horizontal Asymptote when:

as x goes to infinity (or −infinity) the curve approaches some constant value b

Vertical Asymptotes

Vertical Asymptote  

It is a Vertical Asymptote when:

as x approaches some constant value c (from the left or right) then the curve goes towards infinity (or −infinity).

Oblique Asymptotes

Oblique Asymptote  

It is an Oblique Asymptote when:

as x goes to infinity (or −infinity) then the curve goes towards a line y=mx+b

(note: m is not zero as that is a Horizontal Asymptote).

 

Example: (x2-3x)/(2x-2)

The graph of (x2-3x)/(2x-2) has:

  • A vertical asymptote at x=1
  • An oblique asymptote: y=x/2-1
   Asymptote Example

 

These questions will only make sense when you know Rational Expressions: