7. Asymptotics

Calculus 1



Weiqi Zhou

Motiviting Example

Extended Definition

We write if for any , there exists so that holds for any in .

Extended Definition

we say if for any , there exists so that holds for any in .

Signs at Infinity

  • replacing with (resp. )
    we get the definition for the limit of (resp. )

  • replacing with (resp.)
    we get the definition for the limit as (resp. )

The Sequence Case

we write or , if for any ，there exists , so that holds for any

A Useful Criterion

，iff for any with (for all ) that converges to , we always have

Exercises

  • decide whether following statements are correct and why

  • if is bounded，then converges

  • if is unbounded as ，then

  • if ，then

Solution

  • counterexample：

  • counterexample：：
    is unbouned, but it hits infinitely often near
    e.g.,if ，then ，while

  • correct

Comparison

  • suppose

  • if ，then blows up much faster than
    e.g.,:

  • if ，then increases far less than

  • if ，then have comparable speeds of increment. e.g.:

Comparison

  • suppose

  • if ，then decays far less than
    e.g.,:

  • if ，then decays much faster than

  • if ，then have comparable speeds of decaying. e.g.,:

The Little Notation

  • we write if

  • if ，then means decays faster than

  • example：,

The Big Notation

  • we write if there exists ，so that holds for sufficiently large

  • if ，then increase no faster than
    e.g.,：

  • most often is used if have comparable speeds

Summary

  • Definition for the limit being infinity

  • Unbounded the limit equals infinity

  • Comparison