Math 2.0

Approximations and expansions Definition

Little o notation

Builds on Big O notation

Definition

Let ff and gg be real functions defined on an interval [x0,∞)[x_0, \infty). The notation f(x)=o(g(x))f(x) = o(g(x)) as x→∞x \to \infty means that for each ε>0\varepsilon > 0 there is x1≥x0x_1 \ge x_0 with ∣f(x)∣≤ε∣g(x)∣for all x≥x1.|f(x)| \le \varepsilon |g(x)| \quad \text{for all } x \ge x_1.

Remarks

If g(x)≠0g(x) \ne 0 for large xx, then f=o(g)f = o(g) means that f(x)/g(x)→0f(x)/g(x) \to 0. Each f=o(g)f = o(g) is also f=O(g)f = O(g), in the sense of big O notation. For example, x=o(x2)x = o(x^2) and ln⁡x=o(x)\ln x = o(x) as x→∞x \to \infty. Mathlib defines it for filters, as Asymptotics.IsLittleO.

Reading path

To understand this page from the ground up, read these 2 pages in order. The foundations come first.

Show the path as a graph
Big O notation, definition, no evidence levelBig O notationdefinitionLittle o notation, definition, no evidence levelLittle o notationdefinition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Big O notation definition
  2. Little o notation definition

Where this leads

14 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Big_O_notation, Little-o notation
About this page
Tag04TSHB9H8D
URIm2://04TSHB9H8D
Kinddefinition
WritingW0 stub
MSC41A60
DefinesLittle o notation
UsesBig O notation

Markdown · JSON · Edit on GitHub

What this page needs for W1

  • an inventory row: the row of the inventory that the node accounts for (0 of 1)

The writing levels say how complete the text of a page is. They do not measure its truth.

What links here

Used by