The value of `lim_(xrarroo)[(e^(2))/((1+(2)/(x))^(x))]^((x)/(2))` is equal to

Options

A. e
B. `e^(-1)`
C. `e^((1)/(2))`
D. `e^(-(1)/(2))`

Answer

To solve the limit problem \( \lim_{x \to \infty} \left( \frac{e^2}{\left(1 + \frac{2}{x}\right)^x} \right)^{\frac{x}{2}} \), we will follow these steps:

### Step 1: Rewrite the limit
We start with the limit expression:
\[ L = \lim_{x \to \infty} \left( \frac{e^2}{\left(1 + \frac{2}{x}\right)^x} \right)^{\frac{x}{2}} \]

### Step 2: Simplify the expression inside the limit
As \( x \to \infty \), \( \frac{2}{x} \to 0 \). Therefore, we can rewrite the limit as:
\[ L = \lim_{x \to \infty} \left( e^2 \cdot \left(1 + \frac{2}{x}\right)^{-x} \right)^{\frac{x}{2}} \]

### Step 3: Take the natural logarithm
To simplify the limit, we take the natural logarithm:
\[ \ln L = \lim_{x \to \infty} \frac{x}{2} \left( \ln(e^2) - x \ln\left(1 + \frac{2}{x}\right) \right) \]
Since \( \ln(e^2) = 2 \).

### Step 4: Expand \( \ln(1 + \frac{2}{x}) \)
Using the Taylor series expansion for \( \ln(1 + u) \) where \( u = \frac{2}{x} \):
\[ \ln\left(1 + \frac{2}{x}\right) \approx \frac{2}{x} - \frac{1}{2}\left(\frac{2}{x}\right)^2 + O\left(\frac{1}{x^3}\right) \]
Thus, we have:
\[ \ln\left(1 + \frac{2}{x}\right) \approx \frac{2}{x} \quad \text{as } x \to \infty \]

### Step 5: Substitute back into the limit
Now substituting this back into our expression for \( \ln L \):
\[ \ln L = \lim_{x \to \infty} \frac{x}{2} \left( 2 - x \cdot \frac{2}{x} \right) = \lim_{x \to \infty} \frac{x}{2} \left( 2 - 2 \right) = \lim_{x \to \infty} \frac{x}{2} \cdot 0 = 0 \]

### Step 6: Solve for \( L \)
Since \( \ln L = 0 \), we have:
\[ L = e^0 = 1 \]

### Final Answer
Thus, the value of the limit is:
\[ \boxed{1} \]