Let $\alpha$ and $\beta$ be the distinct roots of $a{x^2} + bx + c = 0$, then $\mathop {\lim }\limits_{x \to \alpha } {{1 - \cos \left( {a{x^2} + bx + c} \right)} \over {{{\left( {x - \alpha } \right)}^2}}}$ is equal to

Question

Let $\alpha$ and $\beta$ be the distinct roots of $a{x^2} + bx + c = 0$, then

$\mathop {\lim }\limits_{x \to \alpha } {{1 - \cos \left( {a{x^2} + bx + c} \right)} \over {{{\left( {x - \alpha } \right)}^2}}}$ is equal to

${{{a^2}{{\left( {\alpha - \beta } \right)}^2}} \over 2}$

$ - {{{a^2}{{\left( {\alpha - \beta } \right)}^2}} \over 2}$

${{{{\left( {\alpha - \beta } \right)}^2}} \over 2}$

Solution

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