Mathematics - Limits Question with Solution | TestHub

$$\begin{array}{rl}& \text{ Given limit }=\underset{x\to \alpha }{\lim }\frac{1-\cos a(x-\alpha )(x-\beta )}{(x-\alpha {)}^2}=\underset{x\to \alpha }{\lim }\frac{2{\sin }^2\left(a\frac{(x-\alpha )(x-\beta )}{2}\right)}{(x-\alpha {)}^2}\\ & =\underset{x\to \alpha }{\lim }\frac{2}{(x-\alpha {)}^2}\times \frac{{\sin }^2\left(a\frac{(x-\alpha )(x-\beta )}{2}\right)}{a^2(x-\alpha {)}^2(x-\beta {)}^2}\times \frac{a^2(x-\alpha {)}^2(x-\beta {)}^2}{4}\\ & =\frac{a^2(\alpha -\beta {)}^2}{2}.\end{array}$$