I am new to this forum, so I hope I am proceeding in the correct way. Please excuse any mistakes.
I am trying to prove that the limiting factor of a 2D shape's surface area is a circle, and I have managed to find an equation for this from a regular polygon. In my question, I assume that the maximum perimeter/circumference one can form is 40cm (i.e. if someone has 40cm of string). So:
For a perimeter of 40:
$$
\lim_{n \rightarrow \infty}\frac{200\sin(\frac{2π}{n})}{n\sin^2(\frac{π}{n})}=\frac{400}{π}
$$
I got this value using Wolfram Alpha, which confirmed that the limiting factor for surface area is a circle, since the surface area of a circle with a circumference of 40cm = 400/π.
However, I am unable to prove this formula algebraically. I tried using l'hospital's rule after realising that the limit of the original function was 0/0, but I got nowhere. In fact, the result I got was -400π/0, which was very disappointing after so much working out!
I was wondering if anyone could help me prove this algebraically or otherwise. I am happy that I found an equation that proves what I wanted to, but I am unable to prove Wolfram Alpha's result, which is frustrating.
Again, I am new to this forum, so please let me know if I have made any mistakes so that I can edit my question.
Best Answer
Let $x = \frac {\pi}{n}$
Now we have the more familiar looking:
$\lim_\limits{x\to 0} \frac {200\sin 2x}{(\frac \pi x) \sin^2 x}\\ \lim_\limits{x\to 0} \left(\frac {200}{\pi} \right)\left(\frac {\sin 2x}{\sin x}\right)\left(\frac {x}{\sin x}\right)$