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If α and β are the two different zeros of the quadratic ax² + bx + c = 0, then the value of \nlim_{x→α} \frac{1 - \cos(ax² + bx + c)}{(x - α)²}\n\nis

1. a²/2 (α - β)²
2. 0
3. -a²/2 (α - β)²
4. 1/2 (α - β)²

Correct answer: a²/2 (α - β)²

Solution

The limit evaluates the behavior of the function as x approaches one of its roots, α. By applying L'Hôpital's rule and recognizing that the expression inside the cosine approaches zero, we find that the limit simplifies to a form involving the derivative of the cosine function, leading to the result of a²/2 (α - β)².

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