# User:IssaRice/Little o notation

## Definition

**Definition** (little o near a point). Let and be two functions, and let . We say that is little o of near iff for every there exists such that implies . Some equivalent ways to say the same thing are:

Notation | Comments |
---|---|

is little o of near | This is a point-free notation. |

as | This is in point notation, as the variable appears in the notation. This allows us to define functions anonymously. For example, we can say as ; we didn't even name the functions. As the appearance of the symbol "" suggests, in this notation we think of as a set, namely the set of all functions that are as . |

as | This is in point notation. As the equality symbol suggests, in this notation we think of f as a concrete manifestation of a function that is near . This allows us to algebraically manipulate the expression along with all our other expressions. |

near | This is a point-free notation. As the appearance of the symbol "" suggests, in this notation we think of as a set, namely the set of all functions that are near . In other words, |

near | This is a point-free notation. |

**Definition** (little o at infinity). Let and be two functions. We say that is little o of at positive infinity (or equivalently is little of as ) iff for every real there exists a real number such that for all , if then . We say that is little o of at negative infinity (or equivalently is little of as ) iff for every real there exists a real number such that for all , if then .

**Exercise**. Show that in the definition of little o at positive infinity, "there exists a real number " can be replaced by "there exists a real number ". Show that in the definition of little o at negative infinity, "there exists a real number " can be replaced by "there exists a real number ".

**Exercise**. Can we write just or or or ?

Expand to see solution:

**Exercise**. If we are being a little pedantic, what is wrong with saying " as "?

Expand to see solution:

**Exercise**. Interpret the meaning of .

Expand to see solution:

## Properties

**Proposition**. Let and be two functions, and suppose for all . Then f is little o of g near a if and only if .

**Proposition**. transitivity

**Proposition**. we can replace the in the definition with , right?

**Exercise**. Let be constants. Interpret the statement " as ".

Expand to see solution:

TODO: be careful with universal vs existential quantifiers.

The statement is saying where is some function such that .

Because of the nested little o, we need to expand again and introduce , where so .

Now we verify:

Could have been arbitrary? In other words, could we have said for arbitrary ? To compute the limit we actually used the limit laws, which require that the right hand limit exist. This means that we needed to exist.

## References

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