`lim_(xtooo) [((e)/(1-e))((1)/(e)-(x)/(1+x))]^(x)` is

Options

A. limit does not exist when `a=pi//6`
B. `L=-1` when `a=pi`
C. `L=1` when `a=pi//2`
D. `L=1` when `a=0`

Explanation

To solve the limit \( \lim_{x \to 0} \left[ \frac{e}{1-e} \left( \frac{1}{e} - \frac{x}{1+x} \right) \right]^x \), we will follow these steps:

### Step 1: Simplify the expression inside the limit

We start with the expression inside the limit:

\[ \frac{1}{e} - \frac{x}{1+x} \]

To combine these fractions, we find a common denominator:

\[ \frac{1}{e} - \frac{x}{1+x} = \frac{(1+x) - ex}{e(1+x)} = \frac{1 + x - ex}{e(1+x)} \]

### Step 2: Substitute back into the limit

Now substituting this back into our limit, we have:

\[ \lim_{x \to 0} \left[ \frac{e}{1-e} \cdot \frac{1 + x - ex}{e(1+x)} \right]^x \]

This simplifies to:

\[ \lim_{x \to 0} \left[ \frac{1 + x - ex}{(1-e)(1+x)} \right]^x \]

### Step 3: Analyze the limit as \( x \to 0 \)

As \( x \to 0 \), we can evaluate the expression inside the limit:

\[ 1 + x - ex \to 1 \quad \text{and} \quad 1 + x \to 1 \]

Thus, the expression simplifies to:

\[ \frac{1 + 0 - 0}{(1-e)(1+0)} = \frac{1}{1-e} \]

### Step 4: Substitute this back into the limit

Now we have:

\[ \lim_{x \to 0} \left[ \frac{1}{(1-e)} \right]^x \]

### Step 5: Evaluate the limit

Since \( \frac{1}{(1-e)} \) is a constant, we can evaluate the limit:

\[ \lim_{x \to 0} \left[ \frac{1}{(1-e)} \right]^x = 1 \]

This is because any constant raised to the power of \( 0 \) is \( 1 \).

### Final Result

Thus, the limit is:

\[ \boxed{1} \]