`lim_(x->1) (log_3 3x)^(log_x 3)=`

Answer

To solve the limit \( \lim_{x \to 1} (\log_3 (3x))^{\log_x 3} \), we will follow these steps:

### Step 1: Identify the form of the limit
When we substitute \( x = 1 \) into the expression, we get:
\[ (\log_3 (3 \cdot 1))^{\log_1 3} = (\log_3 3)^{\text{undefined}} = 1^{\infty} \]
This is an indeterminate form, which suggests we can use the exponential limit technique.

**Hint:** Recognize the indeterminate form \( 1^{\infty} \) and prepare to use the limit transformation.

### Step 2: Rewrite the limit using the exponential function
We can express the limit in the form:
\[ y = (\log_3 (3x))^{\log_x 3} \]
Taking the natural logarithm on both sides gives:
\[ \ln y = \log_x 3 \cdot \log_3 (3x) \]
Now we need to find the limit of \( \ln y \) as \( x \to 1 \).

**Hint:** Use logarithmic properties to simplify the expression.

### Step 3: Simplify the expression
Using the change of base formula, we have:
\[ \log_x 3 = \frac{\log_3 3}{\log_3 x} = \frac{1}{\log_3 x} \]
Thus,
\[ \ln y = \frac{1}{\log_3 x} \cdot \log_3 (3x) = \frac{\log_3 3 + \log_3 x}{\log_3 x} = \frac{1 + \log_3 x}{\log_3 x} \]

**Hint:** Simplify the expression further to isolate the limit.

### Step 4: Evaluate the limit
Now we can separate the terms:
\[ \ln y = \frac{1}{\log_3 x} + 1 \]
As \( x \to 1 \), \( \log_3 x \to 0 \), leading to:
\[ \lim_{x \to 1} \ln y = \lim_{x \to 1} \left( \frac{1}{\log_3 x} + 1 \right) \]
The term \( \frac{1}{\log_3 x} \) approaches \( \infty \), so:
\[ \lim_{x \to 1} \ln y = \infty \]

**Hint:** Recognize that \( \ln y \to \infty \) implies \( y \to e^{\infty} \).

### Step 5: Convert back to the original limit
Since \( \ln y \to \infty \), we have:
\[ y \to e^{\infty} = \infty \]
Thus, the original limit evaluates to:
\[ \lim_{x \to 1} (\log_3 (3x))^{\log_x 3} = \infty \]

**Final Result:**
\[ \lim_{x \to 1} (\log_3 (3x))^{\log_x 3} = \infty \]