*Notes to a video lecture on http://www.unizor.com*

__Squeeze Theorem__

In this lecture we present a very important so-called "Squeeze Theorem" for function limits, which we did prove for sequences before.

*Squeeze Theorem*

(also known as

"Pinching Theorem" or

"Sandwich Theorem", or

"Theorem about Two Policemen and a Drunk")

IF, when

*x→r*or

*x→+∞*, or

*x→−∞*,

it is GIVEN that

*AND*

**f(x)→L***AND*

**h(x)→L**

**f(x) ≤ g(x) ≤ h(x)**THEN

(under the same condition of tendency of argument

*x*)

IT IS TRUE that

**g(x)→L**The variant of this theorem is when

**is not a concrete real number, but can be (non-rigorously) positive or negative**

*L**infinity*or (more rigorously) both functions

**and**

*f(x)***can be infinitely increasing or infinitely decreasing.**

*h(x)**Proof*

Let's prove for

*x→r*where

*r*- any real number. The other two conditions, when

*x→+∞*or

*x→−∞*, or cases, when the limit is infinite, will not present any problem as a self-study exercise, they are, generally speaking, similar to the proof below.

We will use

*ε-δ*definition of a limit.

Fix any positive

*ε*. We have to find

*δ*such that, if

**|**

*x−r*| ≤*δ***|**

*g(x)−L*| ≤*ε*For this

*ε*we can find

*δ*such that in

_{1}*δ*-neighborhood

_{1}*x=r*it is true that

**|**

*f(x)−L*| ≤*ε*

*L−ε ≤ f(x) ≤ L+ε*For the same

*ε*we can find

*δ*such that in

_{2}*δ*-neighborhood

_{2}*x=r*it is true that

**|**

*h(x)−L*| ≤*ε*

*L−ε ≤ h(x) ≤ L+ε*Now choose

*δ*equal to a minimum among

*δ*and

_{1}*δ*. Obviously, in

_{2}*δ*-neighborhood

*x=r*it is true that

*L−ε ≤ f(x) ≤ L+ε*

*L−ε ≤ h(x) ≤ L+ε*Therefore, in this

*δ*-neighborhood

*x=r*it is true that

*L−ε ≤ f(x) ≤ g(x) ≤ h(x) ≤ L+ε*Hence,

**|**

*g(x) − L*| ≤*ε*End of Proof.

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