This is indeed the common way most people encounter this. The proof of the difference in the limits for the perimeter vs. the area is in the first answer to the

This is indeed the common way most people encounter this. The proof of the difference in the limits for the perimeter vs. the area is in the first answer to the stack overflow question in the G(^n)P: https://math.stackexchange.com/a/12907 https://math.stackexchange.com/a/12907