Unit 9: The ε–δ definition of a limit
About this unit
The limit of f(x) as x approaches a is L when, for every tolerance ε > 0, some δ > 0 keeps f(x) within ε of L for all x within δ of a (other than a); a jump fails because small ε can never be met.
By the end, you can: Turn 'as close as you like' into a game: for every tolerance ε find a δ that keeps f(x) within ε of L. Win it for 3x + 1 at 2 with δ = ε/3, and see why no δ works at a jump.
Modules
5 modulesLearn
- The tolerance gameArticle · For every ε > 0 there must be a δ > 0 with 0 < |x − a| < δ ⇒ |f(x) − L| < ε; for 3x + 1 at 2, |f(x) − 7| = 3|x − 2| so δ = ε/3.Not started
- Win a tight roundExperiment · Drag ε down to 0.1 or less and find a δ that keeps every output of 3x + 1 near x = 2 within ε of 7.Not started
- When no δ worksArticle · At a jump from 2 to 3, every δ lets in outputs 1 apart, so ε = 0.5 can't be met for any claimed L; a generous ε proves nothing.Not started
- The game in one placeReview · For every ε a δ; δ depends on ε; lines give δ = ε/slope; jumps lose at small ε.Not started