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Squeeze Theorem

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

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Squeeze Theorem

Contents

Theorem

Suppose f(x) is between g(x) and h(x) for all x in a neighborhood of the point 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)\leq f(x) \leq h(x) or h(x)\leq f(x)\leq g(x) for all x in this neighborhood. The two cases are the same up to renaming our functions, so assume without loss of generality that g(x)\leq f(x) \leq h(x).

We must show that for all \varepsilon >0 there is some \delta > 0 for which |x-S|<\delta implies |f(x)-L|<\varepsilon.

Now since \lim_{x\to S}g(x)=\lim_{x\to S}h(x)=L, there must exist \delta_1,\delta_2>0 such that

|x-S|<\delta_1 \Rightarrow |g(x)-L|<\varepsilon \textrm{ and } |x-S|<\delta_2 \Rightarrow |h(x)-L|<\varepsilon.

Now let \delta = \min\{\delta_1,\delta_2\}. If |x-S|<\delta then

-\varepsilon < g(x) - L \leq f(x) - L \leq h(x) - L < \varepsilon.

So |f(x)-L|<\varepsilon. Now by the definition of a limit we get \lim_{x\to S}f(x)=L as desired.

Applications and examples

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See Also

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