The value of where are the distinct roots ax^2 + bx + c = 0 is.

JEE MainMathematicsModerateLimits

The value of
$\lim_{x \rightarrow \alpha}\mspace{2mu}\frac{1 - cos\left( ax^{2} + bx + c \right)}{(x - \alpha)^{2}}$
where $\alpha,\beta$ are the distinct roots ax^2 + bx + c = 0 is.

Show answer and solution

\lim_{x \rightarrow \alpha}\mspace{2mu}\frac{1 - cos\left( ax^{2} + bx + c \right)}{(x - \alpha)^{2}} = \lim_{x \rightarrow \alpha}\mspace{2mu}\frac{2\sin^{2}\frac{ax^{2} + bx + c}{2}}{(x - \alpha)^{2}}

= \lim_{x \rightarrow \alpha}\mspace{2mu}\frac{\frac{\sin^{2}a(x - \alpha)(x - \beta)}{2}}{\left( \frac{a(x - \alpha)(x - \beta)}{2} \right)^{2}} \times \frac{a^{2}}{2}(x - \beta)^{2} = \frac{a^{2}(\alpha - \beta)^{2}}{2}

Source: Grass-G / M-11 Mathematics Paper-2 1Nov.docx

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