`A_(i)=(x-a_(i))/(|x-a_(i)|),i=1,2,...,n," and "a_(1)lta_(2)lta_(3)lt...lta_(n).`
If `1lemlen,minN,` then the value of `L=lim_(xtoa_(m)) (A_(1)A_(2)...A_(n))`is (a) 2 (b) -1 (c) not exist (d) 1

Options

A. 2
B. -1
C. not exist
D. 1

Explanation

To solve the problem, we need to analyze the expression given and evaluate the limit \( L = \lim_{x \to a_m} (A_1 A_2 \ldots A_n) \).

### Step 1: Understanding \( A_i \)
The expression for \( A_i \) is given as:
\[ A_i = \frac{x - a_i}{|x - a_i|} \]
This expression will yield different values based on the position of \( x \) relative to \( a_i \).

### Step 2: Analyzing the left neighborhood of \( a_m \)
Assume \( x \) approaches \( a_m \) from the left (i.e., \( x \to a_m^- \)):
- For \( i < m \) (i.e., \( A_1, A_2, \ldots, A_{m-1} \)):
- \( x - a_i < 0 \) (since \( x < a_m < a_i \))
- Thus, \( A_i = \frac{x - a_i}{|x - a_i|} = -1 \) for \( i = 1, 2, \ldots, m-1 \).

- For \( i = m \):
- \( A_m = \frac{x - a_m}{|x - a_m|} = \frac{x - a_m}{-(x - a_m)} = -1 \).

- For \( i > m \) (i.e., \( A_{m+1}, \ldots, A_n \)):
- \( x - a_i < 0 \) (since \( x < a_m < a_i \))
- Thus, \( A_i = -1 \) for \( i = m+1, \ldots, n \).

### Step 3: Calculating the product \( A_1 A_2 \ldots A_n \) from the left
From the above analysis:
- There are \( m-1 \) terms equal to \(-1\) from \( A_1 \) to \( A_{m-1} \).
- \( A_m = -1 \).
- There are \( n-m \) terms equal to \(-1\) from \( A_{m+1} \) to \( A_n \).

The total number of \(-1\) terms is:
\[ (m-1) + 1 + (n-m) = n \]
Thus, the product is:
\[ A_1 A_2 \ldots A_n = (-1)^n \]

### Step 4: Analyzing the right neighborhood of \( a_m \)
Now, let \( x \) approach \( a_m \) from the right (i.e., \( x \to a_m^+ \)):
- For \( i < m \):
- \( A_i = 1 \) for \( i = 1, 2, \ldots, m-1 \).

- For \( i = m \):
- \( A_m = 1 \).

- For \( i > m \):
- \( A_i = 1 \) for \( i = m+1, \ldots, n \).

The total number of \( 1 \) terms is:
\[ (m-1) + 1 + (n-m) = n \]
Thus, the product is:
\[ A_1 A_2 \ldots A_n = 1^n = 1 \]

### Step 5: Evaluating the limit
We have:
- \( \lim_{x \to a_m^-} (A_1 A_2 \ldots A_n) = (-1)^n \)
- \( \lim_{x \to a_m^+} (A_1 A_2 \ldots A_n) = 1 \)

Since these two limits are not equal, the overall limit \( L \) does not exist.

### Conclusion
Thus, the value of \( L \) is:
\[ \boxed{\text{not exist}} \]