If \( \alpha > \beta > 0 \) are the roots of the equation \( a x^{2}+bx+1=0 \), and \(\displaystyle \lim_{x\to \dfrac{1}{\alpha}} \left(\frac{1-\cos(x^{2}+bx+a)}{2(1-\alpha x)^{2}}\right)^{\dfrac{1}{2}}=\frac{1}{k}\left(\frac{1}{\beta}-\frac{1}{\alpha}\right) \), then \( k \) is equal to:
