{"id":"limit","domain":"analysis","type":"Concept","title":"Limit","slug":"limit","url":"/mathematics/analysis/limit/","summary":"The value a function approaches as its input approaches a point. Foundation of calculus, rigorously defined via epsilon-delta (Cauchy 1821, Weierstrass 1861).","formal_definition":"lim(x→a) f(x)=L: for every ε>0 there exists δ>0 such that 0<|x-a|<δ implies |f(x)-L|<ε.","added_to_codex":"2026-05-27","last_verified":"2026-05-27","sources":[{"tier":1,"citation":"Rudin, W. (1976). Principles of Mathematical Analysis. 3rd ed. McGraw-Hill."},{"tier":1,"citation":"Spivak, M. (2008). Calculus. 4th ed. Publish or Perish."},{"tier":2,"citation":"Robinson, A. (1966). Non-standard Analysis. North-Holland."},{"tier":3,"citation":"Boyer, C.B. (1949). The History of the Calculus and its Conceptual Development. Dover."}],"relationships":[{"type":"underlies","target":"derivative","label":"Derivative"},{"type":"underlies","target":"integral","label":"Integral"},{"type":"uses","target":"function","label":"Function"}],"entry_status":"live","reading_levels_available":["curious","exploring","deep_dive"],"completeness":1.0,"codex_version":"1.4"}