Two Bounds, and Whether the Space Between Them Closes

You'll learnto state both hypotheses of the Squeeze Theorem and check them separately, to build bounds by multiplying an inequality through rather than by quoting one, to notice when a bound is valid on only one side of the target, and to say that the theorem concludes nothing when the two bounds close on different values.

Some functions have no limit and still sit between two functions that do. If the outer two close on the same value, the middle one has nowhere else to go. That is the whole theorem, and the marks are not lost on the idea — they are lost on the checking. Bounds get quoted without being multiplied through, bounds that close on two different values get used anyway, and bounds that only hold to the right of the target get applied to a two-sided limit. Each of those is a picture that does not close, and each one is drawn here.

8 STEPS · 6 QUICK CHECKS · BOTH BOUNDS MUST MEET · v1

bounds that hold near a and close on one value ⇒ the trapped function closes on it too

Before you start

What you're looking at

One graph. Two green curves are the bounds, the shaded strip between them is where the middle function is allowed to be, and the pink curve is a function that has been trapped there.

The question

Does the shaded strip close on a single height as the inputs close in on the target — and does the inequality that produced it actually hold on both sides of the target?

Watch for

Half of these pictures do not close. A strip of fixed width, a strip that exists on only one side of the target, and a band whose two edges never meet are all perfectly true statements that settle nothing.

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