Archimedes and limits

Archimedes computed the circumference of the circle by squeezing the circle between two polygons with n sides obtaining so an approximations of the circumference and especially an approximation of pi. Lets look at his calculation with modern eyes: Writing x=pi/n, the length of the inner polygon is 2n sin(x), the length of the outer polygon is 2n tan(x) and the length of the circle is 2n x. We get the estimate 2n sin(x) ≤ 2n x ≤ 2 n tan(x) . Dividing this by 2n/sin(x) gives 1 ≤ x/sin(x) ≤ 1/cos(x) which is equivalent to cos(x) ≤ sin(x)/x ≤ 1 , from which one can see the fundamental theorem of trigonometry
 
        sin(x)
   lim --------  = 1
          x 
We see that Archimedes picture of approximating the length of the circle from inside and outside by polygons leads to an important result in calculus. The result is important because it allows to get the formulas for the derivatives of the trigonometric functions sin(x) and cos(x) using addition formulas and double angle formulas:
      (1 - cos(x))   =         4 sin2(x/2)    
  lim ------------       lim ----------------------    = 0 
           x                         x
Using this and the fundamental theorem again the limit h to zero can be taken:
  sin(x+h)-sin(x) = sin(x) cos(h) + cos(x) sin(h)-sin(x) 
 ----------------   ------------------------------------ 
        h                           h                 

                  = sin(x) [ cos(h)-1 ] + cos(x) sin(h) 
                    -----------------------------------  -> cos(x) 
                                    h                
In the same way, we get
  cos(x+h)-cos(x) = cos(x) cos(h) - sin(x) sin(h)-cos(x) 
 ----------------   ------------------------------------ 
        h                           h  

                  = cos(x) [ cos(h)-1 ] - sin(x) sin(h)
                   -----------------------------------  -> -sin(x)
                                    h