$$\lim_{\theta \to 0}\frac{\sin\theta}{\theta} = 1$$ The above limit is fundamental to studies of introductory calculus. I know that this limit could be proven by the squeeze theorem and the length of sector, i.e. $$s = r\theta$$ where r is radius and $\theta$ the angle. However, it is claimed that the proof of this limit is circular.
I can't bring myself to agree to that, but apparently the length of sector is a corollary of the limit, which is proven by the inequality $$\cos \theta < \frac{\sin\theta}{\theta} < 1$$ Can anyone point me to other proofs of the limit?