How to prove that $ \underset{\varDelta x\rightarrow 0}{\lim}\frac{\left| MN \right|}{\overset{\frown}{MN}} $

In the derivation of the arc differential formula, why is the limit of the ratio of the length of the line segment between two points to the length of the arc considered to be 1: $$\underset{\varDelta x\rightarrow 0}{\lim}\frac{\left| MN \right|}{\overset{\frown}{MN}}$$How can this be proved?What I've learnt so far is that the limit holds when the function $y = f(x)$ is smooth, are there weaker conditions under which the limit also holds? How can I prove it?