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I've been going over old calculus books to refresh my memory and have mainly been focusing on proofs. One of the things I found interesting was the squeeze theorem, even though since basic calculus i haven't used it much if at all. One of the proofs I know it is used for is $\lim\limits_{x\to0}\frac{\sin x}x$. Bascially the proof consists of making $3$ different area formulas in a sector of the unit circle. The areas of the triangles with height $\sin x$ and $\tan x$ to "squeeze" the area of the sector with angle $x$. I understand and know the proof, my question is more about the theory behind the proof. The thing I don't understand about the proof is how someone went from trying to find $\lim\limits_{x\to0}\frac{\sin x}x$,to using the squeeze theorem in the unit circle. Also why exactly is the Squeeze theorem used instead of some other method?

(On a side note is there any time that the Squeeze theorem is useful for finding limits in upper level math courses or physics courses?)

Edited to hopefully make it more understandable.

Ved Patel
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user2891789
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    I think you are referring to a particular limit here where $x$ goes to zero? The "squeeze" is here that the term $\frac{sinx}{x}$ is "caught" between 1 and 1 and hence it goes to 1. Proofs of this feat are widely available on the internet. – imranfat Apr 09 '15 at 03:19
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    For example, here: http://math.stackexchange.com/a/75151/169852 –  Apr 09 '15 at 05:25

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Perhaps everything that can be defined by using limits, as those are defined by $\varepsilon\text{–}\delta$ techniques, can also be defined by squeezing.

For example, in the very staid conventional first-year calculus course, one writes $$ f'(a) = \lim_{\Delta x\,\to\,0} \frac{f(a+\Delta x) - f(a)}{\Delta x} $$ and takes that to be a definition. But consider the point–slope form of the equation of a line through the point $(a,f(a)){:}$ $$ y-f(a) = m(x-a). $$ If that line crosses from below the graph of $f$ to above as $x$ goes from less than $a$ to more, then $m$ is not too small to be the slope of the graph at that point, i.e. it is not too small to be $f'(a).$ And if that line crosses from above the graph of $f$ to below as $x$ goes from less than $a$ to more, then it is not too large to be $f'(a).$ If there is just one number that is not to large to be $f'(a)$ and also not too small to be $f'(a)$ then that number is $f'(a).$ That is a definition by squeezing rather than by limits.

Another example of squeezing is in the definition of Lebesgue measure, as opposed to Borel measure. Suppose $A\subseteq B\subseteq C$ and $A$ and $C$ are Borel-measurable sets with the same measure. Then $B$ is a Lebesgue-measurable set, also with that same measure.

I am not prepared to give a list of a variety of contexts in advanced mathematics in which squeezing of one sort or another is used, but maybe such a thing can be done.

Michael Hardy
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