# Squeeze theorem and $\frac{\sin x}{x}$

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 the limit as x approaches 0 for $\dfrac{\sin x}{x}$. 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 the limit of $\dfrac{\sin x}{x}$ as it approaches zero, 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.

• 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 9 '15 at 3:19
• For example, here: math.stackexchange.com/a/75151/169852 – Bungo Apr 9 '15 at 5:25