Evaluate `lim_(x to oo) (log_(3)3x)^(log_(x)3).`

Explanation

To evaluate the limit \( \lim_{x \to \infty} (\log_3 (3x))^{\log_x 3} \), we will follow these steps:

### Step 1: Simplify the logarithm
We start with the expression inside the limit:
\[ \log_3 (3x) = \log_3 3 + \log_3 x \]
Using the property of logarithms that states \( \log_a (bc) = \log_a b + \log_a c \).

### Step 2: Substitute back into the limit
Now we can rewrite the limit:
\[ \lim_{x \to \infty} (\log_3 (3x))^{\log_x 3} = \lim_{x \to \infty} (\log_3 3 + \log_3 x)^{\log_x 3} \]

### Step 3: Change the base of the logarithm
Next, we know that \( \log_x 3 = \frac{1}{\log_3 x} \) (using the change of base formula). Thus, we can rewrite the limit as:
\[ \lim_{x \to \infty} (\log_3 3 + \log_3 x)^{\frac{1}{\log_3 x}} \]

### Step 4: Analyze the limit
As \( x \to \infty \), \( \log_3 x \to \infty \). Therefore, we can express the limit as:
\[ \lim_{x \to \infty} \left( \log_3 3 + \log_3 x \right)^{\frac{1}{\log_3 x}} = \lim_{x \to \infty} \left( \log_3 x \left(1 + \frac{\log_3 3}{\log_3 x}\right) \right)^{\frac{1}{\log_3 x}} \]

### Step 5: Simplify further
This can be simplified to:
\[ \lim_{x \to \infty} \left( \log_3 x \right)^{\frac{1}{\log_3 x}} \cdot \left(1 + \frac{\log_3 3}{\log_3 x}\right)^{\frac{1}{\log_3 x}} \]

### Step 6: Evaluate the first part
The first part \( \left( \log_3 x \right)^{\frac{1}{\log_3 x}} \) approaches \( e \) as \( x \to \infty \) because it is of the form \( x^{1/x} \).

### Step 7: Evaluate the second part
The second part \( \left(1 + \frac{\log_3 3}{\log_3 x}\right)^{\frac{1}{\log_3 x}} \) approaches \( 1 \) as \( x \to \infty \).

### Step 8: Combine results
Thus, we have:
\[ \lim_{x \to \infty} \left( \log_3 x \right)^{\frac{1}{\log_3 x}} \cdot \left(1 + \frac{\log_3 3}{\log_3 x}\right)^{\frac{1}{\log_3 x}} = e \cdot 1 = e \]

### Final Answer
Therefore, the limit evaluates to:
\[ \lim_{x \to \infty} (\log_3 (3x))^{\log_x 3} = e \]

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