# Limit

**Type:** Concept  
**Domain:** Analysis  
**Codex URL:** /mathematics/analysis/limit/  
**Entry status:** Live — v1.0 (2026-05-27)

## Summary
The value a function approaches as its input approaches a point, even if never reached exactly. Foundation of calculus.

## Formal (epsilon-delta) definition
lim(x→a) f(x)=L means: for every ε>0, there exists δ>0 such that 0<|x−a|<δ implies |f(x)−L|<ε.

## Historical development
Newton/Leibniz (1660s-80s) used intuitive infinitesimals. Berkeley (1734) criticized these as "ghosts of departed quantities." Cauchy (1821) began rigorization. Weierstrass (1861) completed the epsilon-delta definition.

## Non-standard analysis
Robinson (1966) constructed hyperreal numbers with genuine infinitesimals, rigorously validating Leibniz's original intuitive approach.

## Sources
### Tier 1
- Rudin, W. (1976). *Principles of Mathematical Analysis*. 3rd ed. McGraw-Hill.
- Spivak, M. (2008). *Calculus*. 4th ed. Publish or Perish.
### Tier 2
- Robinson, A. (1966). *Non-standard Analysis*. North-Holland.
### Tier 3
- Boyer, C.B. (1949). *The History of the Calculus and its Conceptual Development*. Dover.

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*Mathematics Codex entry v1.0 — added 2026-05-27 — thecodex.expert/mathematics/*