Let `alpha` and `beta` be the distinct roots of `ax^(2) + bx + c = 0`. Then `underset(x to alpha)(lim) (1 - cos (ax^(2) + bx + c))/((x - alpha)^(2))` equal to

Options

A. `(1)/(2) (alpha - beta)^(2)`
B. `-(a^(2))/(2) (alpha - beta)^(2)`
C. 0
D. `(a^(2))/(2) (alpha - beta)^(2)`

Answer

To solve the limit problem given, we will follow these steps systematically.

### Step 1: Understand the Quadratic Equation
Given the quadratic equation \( ax^2 + bx + c = 0 \), we know that \( \alpha \) and \( \beta \) are its distinct roots. We can express the quadratic in factored form as:
\[ a(x - \alpha)(x - \beta) = 0 \]

### Step 2: Rewrite the Limit Expression
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 can rewrite this as:
\[ ax^2 + bx + c = a(x - \alpha)(x - \beta) \]

### Step 3: Substitute in the Limit
Substituting into the limit, we have:
\[ \lim_{x \to \alpha} \frac{1 - \cos\left(a(x - \alpha)(x - \beta)\right)}{(x - \alpha)^2} \]

### Step 4: Use the Trigonometric Identity
Using the identity \( 1 - \cos \theta = 2 \sin^2\left(\frac{\theta}{2}\right) \), we can rewrite the limit:
\[ \lim_{x \to \alpha} \frac{2 \sin^2\left(\frac{a(x - \alpha)(x - \beta)}{2}\right)}{(x - \alpha)^2} \]

### Step 5: Apply the Limit
Now, we can simplify this limit. As \( x \to \alpha \), \( (x - \alpha) \to 0 \). We can use the fact that \( \sin z \approx z \) when \( z \) is close to 0:
\[ \sin\left(\frac{a(x - \alpha)(x - \beta)}{2}\right) \approx \frac{a(x - \alpha)(x - \beta)}{2} \]
Thus, we have:
\[ \sin^2\left(\frac{a(x - \alpha)(x - \beta)}{2}\right) \approx \left(\frac{a(x - \alpha)(x - \beta)}{2}\right)^2 \]

### Step 6: Substitute Back into the Limit
Substituting this back into our limit expression gives:
\[ \lim_{x \to \alpha} \frac{2 \left(\frac{a(x - \alpha)(x - \beta)}{2}\right)^2}{(x - \alpha)^2} \]
This simplifies to:
\[ \lim_{x \to \alpha} \frac{a^2(x - \alpha)^2(x - \beta)^2}{2(x - \alpha)^2} = \lim_{x \to \alpha} \frac{a^2(x - \beta)^2}{2} \]

### Step 7: Evaluate the Limit
As \( x \to \alpha \), \( (x - \beta) \) approaches \( (\alpha - \beta) \):
\[ \frac{a^2(\alpha - \beta)^2}{2} \]

### Final Result
Thus, the limit evaluates to:
\[ \frac{a^2(\alpha - \beta)^2}{2} \]