Let `g(x)=|(f(x+alpha), f(x+2a), f(x+3alpha)), f(alpha), f(2alpha), f(3alpha),(f\'(alpha),(f\'(2alpha), f\'(3alpha))|`, where alpha is a constant then `Lt_(xrarr0(g(x))/x=` (A) 0 (B) 1 (C) -1 (D) none of these

Options

A. 0
B. 1
C. `-1`
D. none of these

Answer

To solve the problem, we need to evaluate the limit:

\[ \lim_{x \to 0} \frac{g(x)}{x} \]

where

\[ g(x) = \begin{vmatrix} f(x+\alpha) & f(x+2\alpha) & f(x+3\alpha) \\ f(\alpha) & f(2\alpha) & f(3\alpha) \\ f'(\alpha) & f'(2\alpha) & f'(3\alpha) \end{vmatrix} \]

### Step 1: Evaluate \( g(0) \)

First, we need to find \( g(0) \):

\[ g(0) = \begin{vmatrix} f(\alpha) & f(2\alpha) & f(3\alpha) \\ f(\alpha) & f(2\alpha) & f(3\alpha) \\ f'(\alpha) & f'(2\alpha) & f'(3\alpha) \end{vmatrix} \]

### Step 2: Identify the determinant

Notice that the first two rows of the determinant are identical:

\[ \begin{vmatrix} f(\alpha) & f(2\alpha) & f(3\alpha) \\ f(\alpha) & f(2\alpha) & f(3\alpha) \\ f'(\alpha) & f'(2\alpha) & f'(3\alpha) \end{vmatrix} = 0 \]

Since the determinant of a matrix with two identical rows is zero, we have:

\[ g(0) = 0 \]

### Step 3: Apply L'Hôpital's Rule

Since \( g(0) = 0 \), we have an indeterminate form \( \frac{0}{0} \). We can apply L'Hôpital's Rule:

\[ \lim_{x \to 0} \frac{g(x)}{x} = \lim_{x \to 0} g'(x) \]

### Step 4: Differentiate \( g(x) \)

To differentiate \( g(x) \), we use the property of determinants. The derivative of a determinant with respect to \( x \) is given by differentiating each element of the rows with respect to \( x \):

\[ g'(x) = \begin{vmatrix} f'(x+\alpha) & f'(x+2\alpha) & f'(x+3\alpha) \\ 0 & 0 & 0 \\ f'(\alpha) & f'(2\alpha) & f'(3\alpha) \end{vmatrix} \]

### Step 5: Evaluate \( g'(0) \)

Evaluating this at \( x = 0 \):

\[ g'(0) = \begin{vmatrix} f'(\alpha) & f'(2\alpha) & f'(3\alpha) \\ 0 & 0 & 0 \\ f'(\alpha) & f'(2\alpha) & f'(3\alpha) \end{vmatrix} \]

Again, the first row is zero, leading to:

\[ g'(0) = 0 \]

### Step 6: Final Limit Evaluation

Thus, we have:

\[ \lim_{x \to 0} \frac{g(x)}{x} = g'(0) = 0 \]

### Conclusion

The final answer is:

\[ \boxed{0} \]