TITLE: Strictly formal proof of $ \displaystyle \lim_{x \to 0} \frac{\sin(x)}{x} = 1 $.
QUESTION [6 upvotes]: I’m looking for a proof of $ \displaystyle \lim_{x \to 0} \frac{\sin(x)}{x} = 1 $ that does not use other trigonometric functions or any first-order approximation to the sine function. Is this possible? The other proofs that I’ve seen on this website don’t really fit these stringent requirements, so I was hoping to see a different kind of demonstration altogether. Thanks!!

REPLY [3 votes]: I'm assuming that OP uses the geometric (slightly non-rigorous) definition of the trigonometric functions, i.e., $(\cos(t),\sin(t))$ is the point on the unit circle forming angle $t$ with the positive $X$ axis). 
The requested limit (known as the first fundamental limit) can't be computed by any method that relies on the fact that $\sin'(x)=\cos(x)$ (in fact, this limit is required in the proof of that fact), and I think this is the point of this exercise.
Here is a geometric proof, legitimate along these lines.