Evaluate `lim_(ntooo)(pin)^(2//n)`

Answer

To evaluate the limit \( \lim_{n \to \infty} \pi^{\frac{2}{n}} \), we can follow these steps:

### Step 1: Rewrite the limit
Let \( L = \lim_{n \to \infty} \pi^{\frac{2}{n}} \).

### Step 2: Take the natural logarithm
Taking the natural logarithm of both sides, we have:
\[ \log L = \lim_{n \to \infty} \log \left( \pi^{\frac{2}{n}} \right) \]

### Step 3: Apply logarithmic properties
Using the property of logarithms \( \log(a^b) = b \log a \), we can rewrite the expression:
\[ \log L = \lim_{n \to \infty} \frac{2}{n} \log \pi \]

### Step 4: Evaluate the limit
As \( n \to \infty \), the term \( \frac{2}{n} \) approaches 0. Therefore:
\[ \log L = 0 \]

### Step 5: Exponentiate to solve for L
To find \( L \), we exponentiate both sides:
\[ L = e^{\log L} = e^0 = 1 \]

### Conclusion
Thus, the limit is:
\[ \lim_{n \to \infty} \pi^{\frac{2}{n}} = 1 \]