Let \( f \) be a non-negative function in \( [0,1] \) and twice differentiable in \( (0,1) \). If \( \int_{0}^{x} \sqrt{1-(f'(t))^{2}} \, dt = \int_{0}^{x} f(t)\, dt, \; 0 \le x \le 1 \) and \( f(0)=0 \), then \( \displaystyle \lim_{x \to 0} \dfrac{1}{x^{2}} \int_{0}^{x} f(t)\, dt \):
