Squeeze Theorem CalcLens

A function too wild to evaluate directly can still be pinned down, if it is trapped between two functions that agree in the limit. Zoom in and watch the trap close.

Upper bound h(x) =

Squeezed function g(x) =

Lower bound f(x) =

Approaching x =

Window half-width δ = 1

Rescale the vertical axis too

Only the horizontal axis zooms, so the trap visibly narrows. Tick the box to rescale both and watch g keep oscillating all the way down.

The trap closing

Bounds at x = δ, and the gap between them
δ	lower f	upper h	gap

Squeeze Theorem

Three functions. If everywhere near a, and the two outer ones have the same limit L at a, then g has nowhere else to go: its limit is L too.

The shaded region is the trap. Drag δ down and watch it close.

What to watch for

The wild function is never evaluated at a. is undefined at 0 — sin(1/0) means nothing. The theorem never asks for it. It concludes from the neighbours.
The oscillation does not stop; the room for it does. Tick "Rescale the vertical axis" and g wobbles just as violently at δ = 0.001 as at δ = 1. Untick it and the wobbling is crushed flat. Both pictures are true, and students who have only seen the second think the function calms down. It does not.
The hypothesis has to hold. Try the sin(1/x) preset: it is genuinely trapped between −1 and 1, but those bounds never meet, so the gap stays 2 and the theorem concludes nothing. Bounds are not enough — they must agree in the limit.

Putting this in a slide

?g=x%5E2+sin(1%2Fx)&lower=-x%5E2&upper=x%5E2&a=0
   &embed=true&controls=delta,table

Note that a plus sign inside a function must be written %2B, and a slash %2F. Every parameter is in docs/url-api.md.

Keyboard

← →: Change δ (focus the slider first)
Space: Run or stop the zoom
?: This dialog
Esc: Close

Squeeze Theorem CalcLens ?

The trap closing

Squeeze Theorem CalcLens