Formalising a Geometric Idea Using Limits If we choose two points on a circle, they form an arc and a chord. We know that if the angle subtended at the centre of the circle by the two points is small, the arc length is very close to the chord length. I wanted to prove/formalise this idea using limits. I think I’ve found a satisfactory way to do this, and I’ll be posting my method as an answer to this question. Any other approaches/ideas would be appreciated.
 A: 
Consider the circle $C(O,r)$. Take any two points $A$, and $B$ on the circle. The straight line distance $AB$ is the length of the chord in question, and the arc length of the arc $AB$ is the arc length in question. I’ve named them $h$, and $s$, respectively. Let $\angle{AOB}=\theta$. Now, using the law of cosines, $h^2=r^2+r^2-2r^2cos(\theta)$. Simplifying this, $h^2=2r^2(1-cos(\theta)$. This gives, $h(\theta)=r\sqrt{2(1-cos(\theta)}$. Also, $s(\theta)=r\theta$. Now, consider $lim_{\theta \to 0}\frac{h(\theta)}{s(\theta)}$. By cancelling out $r$, we get $\frac{h(\theta)}{s(\theta)}=\frac{\sqrt{2(1-cos(\theta)}}{{\theta}}$. To evaluate the limit of this as $\theta$ tends to $0$, we make use of the fact that $1-cos{\theta}=2sin^2{\frac{\theta}{2}}$. Substituting that in, we get that the limit is equal to $2lim_{\theta \to 0}{\frac{sin(\frac{\theta}{2})}{\theta}}$. This easily evaluates to $1$, and the proposition that as the central angle approaches $0$, the chord length approaches the arc length is proved.
