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Standard Limits to Remove Indeterminate Form

If α and β ar...

Question

If $α$ and $β$ are the roots of the quadratic equation $a x^{2} + b x + c = 0,$ then $lim x \to α \frac{1 - cos (a x^{2} + b x + c)}{(x - α)^{2}}$ is :

\frac{(α - β)^{2}}{2}

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2 (α - β)^{2}

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2 a^{2} (α - β)^{2}

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\frac{a^{2} (α - β)^{2}}{2}

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Solution

The correct option is D $\frac{a^{2} (α - β)^{2}}{2}$
$a x^{2} + b x + c = a (x - α) (x - β)$

$lim x \to α \frac{1 - cos (a x^{2} + b x + c)}{(x - α)^{2}}$

$= lim x \to α \frac{1 - cos (a (x - α) (x - β))}{(x - α)^{2}}$

$= lim x \to α \frac{2 {sin}^{2} (\frac{a (x - α) (x - β)}{2})}{(x - α)^{2}}$

$= lim x \to α \frac{2 {sin}^{2} (\frac{a (x - α) (x - β)}{2})}{\frac{a^{2} (x - α)^{2} (x - β)^{2}}{2^{2}}} \times \frac{a^{2} \times (x - β)^{2}}{4}$

$= 2 \times 1 \times \frac{a^{2} (α - β)^{2}}{4}$

$= \frac{a^{2} (α - β)^{2}}{2}$

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