evaluate `lim_(n->oo)((e^n)/pi)^(1/ n)`

Answer

To evaluate the limit \( \lim_{n \to \infty} \left( \frac{e^n}{\pi} \right)^{\frac{1}{n}} \), we can follow these steps:

### Step-by-Step Solution:

1. **Set up the limit**:
Let \( L = \lim_{n \to \infty} \left( \frac{e^n}{\pi} \right)^{\frac{1}{n}} \).

2. **Take the natural logarithm**:
To simplify the limit, we take the natural logarithm of both sides:
\[ \ln L = \lim_{n \to \infty} \ln \left( \left( \frac{e^n}{\pi} \right)^{\frac{1}{n}} \right) \]

3. **Use the logarithm property**:
Using the property \( \ln(a^b) = b \ln a \), we have:
\[ \ln L = \lim_{n \to \infty} \frac{1}{n} \ln \left( \frac{e^n}{\pi} \right) \]

4. **Expand the logarithm**:
Using the property \( \ln \left( \frac{a}{b} \right) = \ln a - \ln b \):
\[ \ln L = \lim_{n \to \infty} \frac{1}{n} \left( \ln(e^n) - \ln(\pi) \right) \]

5. **Simplify the logarithms**:
We know \( \ln(e^n) = n \) and \( \ln(\pi) \) is a constant:
\[ \ln L = \lim_{n \to \infty} \frac{1}{n} \left( n - \ln(\pi) \right) \]

6. **Separate the limit**:
This can be separated as:
\[ \ln L = \lim_{n \to \infty} \left( 1 - \frac{\ln(\pi)}{n} \right) \]

7. **Evaluate the limit**:
As \( n \to \infty \), \( \frac{\ln(\pi)}{n} \to 0 \):
\[ \ln L = 1 - 0 = 1 \]

8. **Exponentiate to find \( L \)**:
Now, we exponentiate both sides to solve for \( L \):
\[ L = e^1 = e \]

### Final Answer:
Thus, the limit is:
\[ \lim_{n \to \infty} \left( \frac{e^n}{\pi} \right)^{\frac{1}{n}} = e \]