If  $\alpha and\beta$ are the roots of the quadratic equation $a{x}^2+bx+c=0$ then $\underset{x\to \frac{1}{\alpha }}{\lim }\sqrt{\frac{1-\cos (c{x}^2+bx+a)}{2{(1-\alpha x)}^2}}=$

$\begin{array}{l}a{x}^2+bx+c=0\Rightarrow roots\alpha ,\beta \\ C{x}^2+bx+a=0\\ \Rightarrow roots\frac{1}{\alpha },\frac{1}{\beta }\\ =\stackrel{Lt}{x\to }\frac{1}{\alpha }\sqrt{\frac{1-\cos \left(c(x-\frac{1}{\alpha })(x-\frac{1}{\beta })\right)}{2{(x-\frac{1}{\alpha })}^2.{\alpha }^2}}\\ =\stackrel{Lt}{x\to }\frac{1}{\alpha }\sqrt{\frac{1}{2}\frac{1}{2{\alpha }^2}{C}^2{(x-\frac{1}{\beta })}^2}\\ =\frac{1}{2}\frac{\left|c\right|}{\left|\alpha \right|}.|\frac{1}{\alpha }-\frac{1}{\beta }|\end{array}$