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Asymptote

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

Asymptote

Types

There are three types: horizontal, vertical and oblique:

Asymptote Types

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

Asymptote Crossing And 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

Horizontal Asymptotes

Horizontal Asymptote  

It is a Horizontal Asymptote when:

as x goes to infinity (or to -infinity) then the curve approaches some fixed 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 to -infinity) then the curve goes towards a line defined by y=mx+b (note: m is not zero as that would be horizontal).

 

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