If C satisfies the equation `lim_(xto oo)((x+c)/(x-c))^(x)=4` then `|(e^(c))/2|` is ………

Answer

To solve the problem, we need to evaluate the limit given in the equation and find the value of \(|(e^c)/2|\).

### Step-by-Step Solution

1. **Start with the Limit Expression**:
We have the limit:
\[ \lim_{x \to \infty} \left(\frac{x+c}{x-c}\right)^x = 4 \]

2. **Simplify the Fraction**:
Rewrite the fraction inside the limit:
\[ \frac{x+c}{x-c} = \frac{x(1 + \frac{c}{x})}{x(1 - \frac{c}{x})} = \frac{1 + \frac{c}{x}}{1 - \frac{c}{x}} \]

3. **Rewrite the Limit**:
Substitute this back into the limit:
\[ \lim_{x \to \infty} \left(\frac{1 + \frac{c}{x}}{1 - \frac{c}{x}}\right)^x \]

4. **Identify the Indeterminate Form**:
As \(x \to \infty\), both \(\frac{c}{x} \to 0\), leading to the form \(1^\infty\).

5. **Use the Exponential Limit**:
We can rewrite this limit using the exponential function:
\[ \lim_{x \to \infty} \left(\frac{1 + \frac{c}{x}}{1 - \frac{c}{x}}\right)^x = e^{\lim_{x \to \infty} x \left(\frac{1 + \frac{c}{x}}{1 - \frac{c}{x}} - 1\right)} \]

6. **Simplify the Expression Inside the Limit**:
We need to simplify:
\[ \frac{1 + \frac{c}{x}}{1 - \frac{c}{x}} - 1 = \frac{(1 + \frac{c}{x}) - (1 - \frac{c}{x})}{1 - \frac{c}{x}} = \frac{\frac{2c}{x}}{1 - \frac{c}{x}} \approx \frac{2c}{x} \text{ as } x \to \infty \]

7. **Evaluate the Limit**:
Now, substituting back:
\[ \lim_{x \to \infty} x \cdot \frac{2c}{x} = 2c \]
Thus, we have:
\[ e^{2c} = 4 \]

8. **Solve for \(c\)**:
Taking the natural logarithm:
\[ 2c = \ln(4) \]
Therefore:
\[ c = \frac{\ln(4)}{2} = \ln(2) \]

9. **Find \(|(e^c)/2|\)**:
Now we need to calculate:
\[ \left|\frac{e^c}{2}\right| = \left|\frac{e^{\ln(2)}}{2}\right| = \left|\frac{2}{2}\right| = 1 \]

### Final Answer
\[ |(e^c)/2| = 1 \]