The Precision Game

Intro to Proofs · Axiom Academy

A challenger names how close your output must land; you answer with how tightly to pin the input — and you have to win for every demand. What does it mean to be "as close as you like"? "The limit of f at a is L " sounds like a story about getting close. To make it a theorem you can prove, mathematicians turned it into a game between two players. A challenger names a tolerance — a band around L your output must land inside. You answer with a precision — a band around a . You win the round if pinning the input to within of a forces the output to within of L . The catch: you must win for every the challenger throws, no matter how small. Watch one full exchange. The challenger keeps shrinking the green -band on the output axis . Each time, a blue -band narrows on the input axis until the slice of the curve above it — the image of f — drops entirely inside . Here f(x)=2x+1 near a=3 , where L=7 . Every time the green band tightens, the blue one answers. That two-step — you name , I answer — is the limit. Your turn: answer the challenge The challenger has set a tolerance (the green band). Now you pick — slide it down, or drag the handle on the input axis. The orange strip is the image of f over your interval . While that strip pokes outside the green band you've lost the round; tighten until it fits, and the win latches. Move the slider to see the demand change. For this straight line the rule is exact: the image is tall, so it fits inside precisely when .

This is the written version of the interactive lesson above. See the full Intro to Proofs course.