AP Calculus AB and BC glossary

Squeeze theorem

Also called: Sandwich theorem, Pinching theorem

The squeeze theorem says that if a function is trapped between two others near a point, and those two share the same limit there, the trapped function must have that limit too. It is the standard way to evaluate limits that resist algebra, such as x squared times sine of one over x.

g(x) \le f(x) \le h(x) \ \text{and} \ \lim_{x \to a} g(x) = \lim_{x \to a} h(x) = L \implies \lim_{x \to a} f(x) = L

The inequality only needs to hold near $a$ , not everywhere, and it does not need to hold at $a$ itself. The theorem is useful precisely when $f$ oscillates too wildly for direct methods but is bounded by something tame.

The classic application uses $-1 \le \sin\left(\frac{1}{x}\right) \le 1$ , so $-x^2 \le x^2 \sin\left(\frac{1}{x}\right) \le x^2$ . Both bounds go to 0 as $x \to 0$ , so the middle function does too.

The mistake

Choosing bounds whose limits are different. If $g$ and $h$ do not share the same limit, the theorem says nothing at all.

Appears in: Unit 1: Limits and Continuity