If we consider the limit case of eg. shrinking a coin, surely at Volume=0 then the diameter of the coin will be 0? Since a coin of volume zero cannot have a diameter (to have a diameter it needs to have a volume of more than zero)?

This random question arose when studying the divergence of a vector field which is defined over $\beta \rightarrow 0$ (the maximum linear dimension or diameter of the "control volume" surrounding a point $P=(x,y,z)$ in a region $R$) and strictly not the volume of the area surrounding $P$.