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

Options

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

Answer

To solve the limit \( \lim_{x \to \infty} \left( \frac{e}{\left(1 + \frac{1}{x}\right)^x} \right)^x \), we will proceed step by step.

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

### Step 2: Analyze the denominator
Next, we need to analyze the term \( \left(1 + \frac{1}{x}\right)^{x^2} \). Using the property that \( \lim_{x \to \infty} \left(1 + \frac{1}{x}\right)^x = e \), we can express:
\[ \left(1 + \frac{1}{x}\right)^{x} \approx e \quad \text{as } x \to \infty \]
Thus,
\[ \left(1 + \frac{1}{x}\right)^{x^2} = \left(\left(1 + \frac{1}{x}\right)^{x}\right)^{x} \approx e^x \]

### Step 3: Substitute back into the limit
Now substituting this back into our limit, we have:
\[ \lim_{x \to \infty} \frac{e^x}{e^x} = \lim_{x \to \infty} 1 = 1 \]

### Step 4: Final expression
Thus, the value of the limit is:
\[ \lim_{x \to \infty} \left( \frac{e}{\left(1 + \frac{1}{x}\right)^x} \right)^x = 1 \]

### Conclusion
Therefore, the final answer is:
\[ \boxed{1} \]