Squeeze Theorem

This is an AoPSWiki Word of the Week for May 4-11

The Squeeze Theorem (also called the Sandwich Theorem or the Squeeze Play Theorem) is a relatively simple theorem that deals with calculus, specifically limits.

Squeeze Theorem

Theorem

Suppose $f(x)$ is between $g(x)$ and $h(x)$ for all $x$ in the neighborhood of $S$ . If $g$ and $h$ approach some common limit L as $x$ approaches $S$ , then $\lim_{x\to S}f(x)=L$ .

Proof

If $f(x)$ is between $g(x)$ and $h(x)$ for all $x$ in the neighborhood of $S$ , then either $g(x)<f(x)<h(x)$ or $h(x)<f(x)<g(x)$ for all $x$ in the neighborhood of $S$ . Since the second case is basically the first case, we just need to prove the first case.

If $g(x)$ increases to $L$ , then $f(x)$ goes to either $L$ or $M$ , where $M>L$ . If $h(x)$ decreases to $L$ , then $f(x)$ goes to either $L$ or $N$ , where $N<L$ . Since $f(x)$ can't go to $M$ or $N$ , then $f(x)$ must go to $L$ . Therefore, $\lim_{x\to S}f(x)=L$ .

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Squeeze Theorem

Theorem

Proof

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