What is Infinity?

A set has infinite cardinality if it can be mapped one-to-one onto a proper subset of itself. This is not the use of infinity in calculus.

In Calculus, we use "infinity" in 3 ways.

Interval notation:
The symbols #oo# (respectively #-oo#) are used to indicate that an interval does not have a right (respectively left) endpoint.

The interval #(2,oo)# is the same as the set #{x|x > 2}#

Infinite Limits

If a limit fails to exist because as #x# approaches #a#, the values of #f(x)# increase without bound, then we write #lim_(xrarra)f(x) = oo#

Note that: the phrase "without bound" is significant. The nubers:
#1/2, 3/4, 7/8, 15/16, 31/32, 63/64 . . . # are increasing, but bounded above. (They never get to or pass #1#.)

Limits at Infinity

The phrase "the limit at infinity" is used to indicate that we have asked what happens to #f(x)# as #x# increases without bound.

Examples include

The limit as #x# increases without bound of #x^2# does not exists because, as #x# increases without bound, #x^2# also increases without bound.

This is written #lim_(xrarr00)x^2 = oo# and we often read it

"The limit as #x# goes to infinity, of #x^2# is infinity"

The limit #lim_(xrarroo) 1/x = 0# indicates that,

as #x# increases without bound, #1/x# approaches #0#.

#bb +-# Infinity and limits

Consider the set of Real numbers #RR#, often pictured as a line with negative numbers on the left and positive numbers on the right. We can add two points called #+oo# and #-oo# that do not quite work as numbers, but have the following property:

#AA x in RR, -oo < x < +oo#

Then we can write #lim_(x->+oo)# to mean the limit as #x# gets more and more positive without upper bound and #lim_(x->-oo)# to mean the limit as #x# gets more and more negative without lower bound.

We can also write expressions like:

#lim_(x->0+) 1/x = +oo#

#lim_(x->0-) 1/x = -oo#

...meaning that the value of #1/x# increases or decreases without bound as #x# approaches #0# from the 'right' or 'left'.

So in these contexts #+-oo# are really shorthand to express conditions or results of limiting processes.

Infinity as a completion of #RR# or #CC#

The projective line #RR_oo# and Riemann sphere #CC_oo# are formed by adding a single point called #oo# to #RR# or #CC# - the "point at infinity".

We can then extend the definition of functions like #f(z) = (az+b)/(cz+d)# to be continuous and well defined on the whole of #RR_oo# or #CC_oo#. These Möbius transformations work particularly well on #C_oo#, where they map circles to circles.

Infinity in Set Theory

The size (Cardinality) of the set of integers is infinite, known as countable infinity. Georg Cantor found that the number of Real numbers is strictly larger than this countable infinity. In set theory there are a whole plethora of infinities of increasing sizes.

Infinity as a number

Can we actually treat infinities as numbers? Yes, but things don't work as you expect all of the time. For example, we might happily say #1/oo = 0# and #1/0 = oo#, but what is the value of #0 * oo ?#

There are number systems which include infinities and infinitesimals (infinitely small numbers). These provide an intuitive picture of the results of limit processes such as differentiation and can be treated rigorously, but there are quite a few pitfalls to avoid.