# Asymptote

An asymptote is a **line** that a curve approaches, as it heads towards infinity:

## Types

There are three types: horizontal, vertical and oblique:

The direction can also be negative:

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

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

It is a Horizontal Asymptote when:

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

## Vertical Asymptotes

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

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: (x^{2}−3x)/(2x−2)

The graph of (x^{2}-3x)/(2x-2) has:

- A vertical asymptote at
**x=1** - An oblique asymptote:
**y=x/2 − 1**

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