Let $\mathrm{\alpha }\text{ and }\mathrm{\beta }$ be the distinct roots of ${\mathrm{ax}}^2+\mathrm{bx}+\mathrm{c}=0$ then the value of $\underset{\mathrm{x}\to \mathrm{\alpha }}{lim} \frac{1-\cos \left({\mathrm{ax}}^2+\mathrm{bx}+\mathrm{c}\right)}{(\mathrm{x}-\mathrm{\alpha }{)}^2}$ is

$\underset{\mathrm{x}\to \mathrm{\alpha }}{lim} \frac{1-\cos \left({\mathrm{ax}}^2+\mathrm{bx}+\mathrm{c}\right)}{(\mathrm{x}-\mathrm{\alpha }{)}^2}$

as $\mathrm{\alpha }\text{ and }\mathrm{\beta }$ are two distinct roots.

$\begin{array}{l}\therefore {\mathrm{ax}}^2+\mathrm{bx}+\mathrm{c}=\mathrm{a}(\mathrm{x}-\mathrm{\alpha })(\mathrm{x}-\mathrm{\beta })\\ \text{ i.e. }\mathrm{\alpha },\mathrm{\beta }=\frac{-\mathrm{b}\pm \sqrt{{\mathrm{b}}^2-4\mathrm{ac}}}{2\mathrm{a}}\\ =\underset{\mathrm{x}\to \mathrm{\alpha }}{lim} \frac{1-\cos \left[\right(\mathrm{x}-\mathrm{\alpha }\left)\right(\mathrm{x}-\mathrm{\beta }\left)\mathrm{a}\right]}{(\mathrm{x}-\mathrm{u}{)}^2}\end{array}$

$\begin{array}{l}=\underset{\mathrm{x}\to \mathrm{\alpha }}{lim} \frac{2{\sin }^2\frac{\left[\right(\mathrm{x}-\mathrm{\alpha }\left)\right(\mathrm{x}-\mathrm{\beta }\left)\mathrm{a}\right]}{2}}{(\mathrm{x}-\mathrm{\alpha }{)}^2}\\ =\underset{\mathrm{x}\to \mathrm{\alpha }}{lim} \frac{2{\sin }^2\frac{\left[\right(\mathrm{x}-\mathrm{\alpha }\left)\right(\mathrm{x}-\mathrm{\beta }\left)\mathrm{a}\right]}{2}}{{\left[\frac{(\mathrm{x}-\mathrm{\alpha })(\mathrm{x}-\mathrm{\beta })\mathrm{a}}{2}\right]}^2}{\mathrm{a}}^2{\left(\frac{\mathrm{x}-\mathrm{\beta }}{4}\right)}^2\\ =\underset{\mathrm{x}\to \mathrm{\alpha }}{lim} \frac{2}{4}{\mathrm{a}}^2(\mathrm{x}-\mathrm{\beta }{)}^2\\ =\frac{{\mathrm{a}}^2}{2}(\mathrm{\alpha }-\mathrm{\beta }{)}^2\end{array}$