Question

Let $ \alpha $ and $ \beta $ be the distinct roots of $ ax^2 + bx + c = 0 $, then $ \lim_{x \to \alpha} \frac{1 - \cos(ax^2 + bx + c)}{(x - \alpha)^2} $ is equal to:

• A. \( \frac{a^2(\alpha - \beta)^2}{2} \)
• B. \( \frac{(\alpha - \beta)^2}{2} \)
• C. \( \frac{-a^2(\alpha - \beta)^2}{2} \)
• D. 0

Hint:

When dealing with limits involving trigonometric functions, use Taylor series expansion for the function around the point where the limit is taken, and apply the first two terms to simplify the expression.

Answer 1

The Correct Option is A

We are given that \( \alpha \) and \( \beta \) are distinct roots of the quadratic equation \( ax^2 + bx + c = 0 \), which implies: \[ a\alpha^2 + b\alpha + c = 0 \quad \text{and} \quad a\beta^2 + b\beta + c = 0. \] We need to evaluate the limit: \[ \lim_{x \to \alpha} \frac{1 - \cos(ax^2 + bx + c)}{(x - \alpha)^2}. \] Since \( ax^2 + bx + c = 0 \) at \( x = \alpha \), we have: \[ \lim_{x \to \alpha} \frac{1 - \cos(0)}{(x - \alpha)^2} = \lim_{x \to \alpha} \frac{1 - 1}{(x - \alpha)^2} = 0. \] Now, we expand \( ax^2 + bx + c \) around \( x = \alpha \) using a Taylor expansion. Let \( f(x) = ax^2 + bx + c \). Near \( x = \alpha \), we use the first two terms of the expansion: \[ f(x) = f(\alpha) + f'(\alpha)(x - \alpha) + \cdots \] where \( f(\alpha) = 0 \) (since \( \alpha \) is a root), and \[ f'(x) = 2ax + b. \] Thus, at \( x = \alpha \), \[ f'(\alpha) = 2a\alpha + b. \] Now, we have: \[ f(x) = (2a\alpha + b)(x - \alpha) + O((x - \alpha)^2). \] Using this expansion in \( \cos(f(x)) \), we get: \[ \cos(f(x)) \approx 1 - \frac{(f(x))^2}{2}. \] Substituting \( f(x) = (2a\alpha + b)(x - \alpha) \), we get: \[ \cos(f(x)) \approx 1 - \frac{((2a\alpha + b)(x - \alpha))^2}{2}. \] Thus: \[ 1 - \cos(f(x)) \approx \frac{((2a\alpha + b)(x - \alpha))^2}{2}. \] Now, substituting this into the original limit expression, we have: \[ \frac{1 - \cos(f(x))}{(x - \alpha)^2} \approx \frac{((2a\alpha + b)(x - \alpha))^2}{2(x - \alpha)^2} = \frac{(2a\alpha + b)^2}{2}. \] Using the fact that \( \alpha \) and \( \beta \) are distinct roots, we conclude that the correct value for the limit is: \[ \frac{a^2(\alpha - \beta)^2{2}}. \] Thus, the correct answer is \( (A) \).