"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.