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.
