# Tends-to relation

This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.

The notation x→a says that the state described by a variable or index approaches the target a.

mko-tends-to-relation · concept · v0.8.0



## Mathematical statement

The relation x→a is directional language used in limits. By itself it does not say whether function values converge, nor does it require x eventually to equal a.





## Representative formula (exact TeX)

x\to a





## Assumptions


- A mathematical structure supporting neighborhoods and approach, such as a metric or topology, has been specified.





## Symbol scope


- x — varying expression or variable
- a — target of approach
- → — tends-to relation





## Intuition, definition, and misconceptions

Approach describes a process or relation rather than a single substitution. The value x may come arbitrarily close to a without being equal to a at every stage under discussion.

A complete function-limit statement must also specify the target of f(x), the order of quantifiers, and neighborhood conditions. This object supplies only the approach component.

Mathlib expresses convergence of a function relative to source and target filters as Filter.Tendsto f l₁ l₂, which is definitionally equivalent to Filter.map f l₁≤l₂.

A common mistake is to read x→a as x=a, or to claim that a bare arrow already equals a complete Filter.Tendsto or epsilon-delta definition before source and target filters are specified.





## Adopted derivation guide

[Tends-to relation: x→a is not x=a, and does not guarantee that function values converge](https://ocme.evemisslab.com/en/derivations/derivation-mko-tends-to-relation-01/)The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.





## Prerequisites


- [Function mapping](https://ocme.evemisslab.com/en/mko/mko-function-mapping/) — A function limit studies the behavior of mapped values as inputs approach a target.





## Verification boundary

Lean formalizes the core filter semantics of Filter.Tendsto. The bare notation x→a still needs a function and source and target filters, and this evidence does not prove a limit for any particular function.

formalized_filter_tendsto_core


- [evidence-sha256-38b848281997cbb1a2c1ea5559c6937dbb67ee03ac0be338756320a99e0f8b1d](https://ocme.evemisslab.com/data/evidence/38b848281997cbb1a2c1ea5559c6937dbb67ee03ac0be338756320a99e0f8b1d.json)





## Next obligation

Create objects for commonly used source and target filters such as atTop, nhds, and nhdsWithin, then connect them separately to sequence limits and pointwise function limits.





## Same-source structured data

This page and the interactive view use the same object and existing language edition; this adds no proof or human review.

[Canonical MKO JSON](https://ocme.evemisslab.com/data/mko/tends-to-relation.json) · [Plain Markdown](https://ocme.evemisslab.com/en/mko/mko-tends-to-relation/index.md)