The Squeeze Theorem

Calculus 1 · Axiom Academy

Your GPS watch can't pin your exact pace — only a low guess and a high guess. But when those two guesses pinch together, your true pace has nowhere to hide. You're on a run wearing a GPS tracker. Buildings and trees scramble the signal, so it never reports one exact speed — it reports a min pace and a max pace you must be between. The trick: when the min and the max both close in on the same number, your real pace is squeezed onto it too. Drag Tightness to close the gap between the min and max pace. As the band pinches toward 5.0 mph , the actual pace caught inside has nowhere left to go — hit Run to send the runner through and watch it land on 5.0. The signal that won't sit still Zoom in on the raw GPS error right as you start moving. It thrashes faster and faster — but it's pinned between -x^2 and +x^2 , two parabolas that both crash to 0 . Drag x toward 0 and watch the wild error get crushed to zero. Now you set the trap. Make the upper and lower bounds as loose or as lopsided as you like, then drag Close the gap . No matter how wild the start, both bounds converge to 5.0 mph — so the pace squeezed between them is forced there too. That's the whole theorem, in your hands.

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