Hi Leonardommarques,

"What is the limit of this curve/function... "

I assume that by the

**curve/function** you refer to

**the sum of perimeters of all inside semicircles**. Below is how I understand your prof's trick.

*May it help*
Assume there are totally n inside semicircles. For an inside semicircle, its diameter is 2/n; so its radius is 1/n and perimeter is

**pi/n**.

When n is very large or even infinite, the limit of

**pi/n** approaches 0.

But, the interesting part is that,

**regardless of the value of n, the sum of perimeters of inside semicircles is always (pi/n)n=pi.** So I'm not quite sure about what kind of limit your prof. questioned.

Another path against the conclusion pi=2 is through analyzing the estimation error.

For an inside semicircle, its perimeter is pi/n and its diameter (i.e., segment to which the perimeter approaches when n is large) is 2/n. When using the diameter to approximate the perimeter, the estimation error is (pi-2)/n. Though for one inside semicircle this error approaches to 0, the sum of the estimation errors of all inside semicircles is [(pi-2)/n]n=pi-2. Now it's pretty clear. It's okay to say that the sum of perimeters of all inside semicircles is

**approximately** equal to 2. But we cannot directly equal it to 2. Combining the estimation error, the sum of perimeters of all inside semicircles should be 2+(pi-2)=pi.