The points A, B and C lie on a circle with centre O, as shown in the diagram - HSC - SSCE Mathematics Extension 1 - Question 12 - 2017 - Paper 1

To evaluate the limit:

1. We use L'Hôpital's rule, as we have an indeterminate form of 0/0:

   • [ ext{Let } L = ext{lim}_{x → 0} rac{1 - ext{cos}(2πx)}{x^2} ]

2. Differentiate the numerator and the denominator:

   • Derivative of the numerator:

   rac{d}{dx}(1 - ext{cos}(2πx)) = 2π ext{sin}(2πx)

   • Derivative of the denominator:
   • rac{d}{dx}(x^2) = 2x

3. Applying L'Hôpital's rule gives:

   • [ L = ext{lim}{x → 0} rac{2π ext{sin}(2πx)}{2x} = ext{lim}{x → 0} rac{ ext{π sin}(2πx)}{x} = ext{π(2π) = 4π^2}]

4. Hence, the limit evaluates to:

   $L = 2 ext{π}$.

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