Racing to Zero

Calculus 1 · Axiom Academy

A quantity that shrinks without limit — and never quite arrives. This is the idea every limit in calculus is built on. Take a fraction and make the bottom enormous. The value gets tiny — and the bigger the bottom, the tinier it gets, with no smallest value it ever settles on. Watch one quantity do exactly that, then drive two more yourself. Here is y = 1/x . Press play and a marker rides the curve as x grows: it dives toward the x-axis, and the running value 1/x shrinks toward 0 — getting closer and closer, but never landing on it. The value never reaches zero — it just keeps getting closer. That "closer and closer, forever" is the whole idea. Same idea, now under your control. Slide the handle to grow n; each bar has height 1/n — the value of the fraction. The more you grow n, the shorter every bar, and the smaller the gap left to the floor at 0. The value and its distance to zero are the same number — and you can drive both as close to 0 as you like. Beat any target — but never hit zero Name a tiny target band just above 0 by dragging (epsilon) down. The toy finds how far out x must go before the curve 1/x slips below your band — and it always can. No matter how small you make , there is an x that beats it. Yet 1/x is never 0. Once x passes 1/ , the curve stays inside your band — for good. The band can shrink forever; an x to clear it always exists.

This is the written version of the interactive lesson above. See the full Calculus 1 course.