Limit of a Family of Curves
A semicircle of radius 1 and its diameter are approximated by two families of curves, which converge to the arc and the diameter in the limit. The areas between the jagged lines and the limits tend to 0 as grows. The sum of the lengths of the jagged lines of the first family tends to the length of the arc of the circle. But the sum of the lengths of the broken lines of the second family does not tend to the length of the diameter.