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So I was working on drawing graph of sin(x)/x, and realised that I need to study circle a little bit more. Also drawing graph of said function requires knowledge of limits.

I tried to find perimeter of circle using chord lengths that grow smaller with each step. And arrived at a formula something like, √(2-√(4–chord ²)) , so you can take chord1=√2, and chord2=√(2-√2) and so on you can get chord(n). On paper one can see that, the number of sectors increases twice each step. So chord1 we had 2² sectors. For chord 2 we have twice as many sectors, ie 2³. And so on.

Now the limit (n tends to infinity) of final expression should give us the perimeter of unit circle precisely. I don't want to call it 2πr outta thin air. But all one can see, that this perimeter is always a multiple of r and that constant, if your replace unity by r. And likewise if you calculate area, the resulting expression has same nested radical thing, but with a factor r²/2.

What I'm looking for is a way to reduce it in forms that doesnt land me having an indeterminate forms of limits. 2^n tends to infinity, while the nested radical tends to zero. It is clear that limits are insufficient here. The knowledge of point where any sequence reaches is simple unable to explain how it reaches that point. So 1/x and 1/2+x and 1/x² all reach zero when x tends to infinity, but the way they reach zero is not explained by limits. So that large nested radical thing is seemingly confusing me. Online I looked for and found everyone retreats back to sine, cosine ,2/pi etc. While sine and cosine are more fundamental(older) if we consider their triangular/ratio definitions, I want to do it purely from the expression I wrote in heading, without using outside elements like sine? Is this possible to manipulate it algebraically?

I understood the sine method tho, sinx/x is not a foriegn element but a direct consequence of the equation x²+y²=1. I just want to manipulate the expression without retreating back to sine and x²+y²=1.