Limit - Revision history

Lfox: Created page with "The '''limit''' (of a Cauchy procedure) is the target which a Cauchy procedure approaches. [TODO ugh] == Examples == === Planes are the limit of increasingly large spheres === Work with 3 dimensional rectilinear coordinates, and consider some sequence of spheres, $C_1, C_2, \cdots$ on the plane, where the sphere $C_n$ is centered at $(0, 0, n)$ and has radius $n$ . Near the origin $(0,0,0)$ , the sphere $C_n$
2024-07-03T17:25:08Z

Created page with "The '''limit''' (of a Cauchy procedure) is the target which a Cauchy procedure approaches. [TODO ugh] == Examples == === Planes are the limit of increasingly large spheres === Work with 3 dimensional rectilinear coordinates, and consider some sequence of spheres, <math>C_1, C_2, \cdots </math> on the plane, where the sphere <math>C_n</math> is centered at <math>(0, 0, n) </math> and has radius <math>n</math>. Near the origin <math>(0,0,0)</math>, the sphere <math>C_n</..."

New page
The '''limit''' (of a Cauchy procedure) is the target which a Cauchy procedure approaches. [TODO ugh]

== Examples ==

=== Planes are the limit of increasingly large spheres ===
Work with 3 dimensional rectilinear coordinates, and consider some sequence of spheres, <math>C_1, C_2, \cdots </math> on the plane, where the sphere <math>C_n</math> is centered at <math>(0, 0, n) </math> and has radius <math>n</math>. Near the origin <math>(0,0,0)</math>, the sphere <math>C_n</math> will look increasingly similar to the xy plane <math>P := \{ (x,y,z) : z =0 \} </math>, as <math>n</math> gets larger. For any given context, there exists some <math>N</math> such that when <math>n\geq N</math>, the sphere <math>C_n</math> is ''indistinguishable'' from the plane. We say [TODO should OM say?] that the limit of the sequence <math>\{ C_n \}_{n=1}^\infty</math> is <math>P</math>.

This example is very practical; it is why we may think of the Earth as being flat in most everyday contexts. The Earth is a sphere, but it has a radius which is so large that near a point it can be identified as a plane (or to state it more confusingly, it ''is'' a plane).

=== Turing Machines are the limit of increasingly robust computer programs [TODO] ===
Turing Machines are an idealization of computer programs [TODO no no no no no no no no no no no that's not the right way to think about it!] in the "limit" where the runtime increases. blech