The value of `"lt"_(x to 1)[log_(5) 5x]^(log_(x) 5)` is:

Options

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

Answer

To solve the limit \( \lim_{x \to 1} \left( \log_{5}(5x) \right)^{\log_{x}(5)} \), we can follow these steps:

### Step 1: Rewrite the expression
We start by rewriting the limit expression:
\[ \lim_{x \to 1} \left( \log_{5}(5x) \right)^{\log_{x}(5)} \]

### Step 2: Simplify the logarithm
Using the property of logarithms, we can simplify \( \log_{5}(5x) \):
\[ \log_{5}(5x) = \log_{5}(5) + \log_{5}(x) = 1 + \log_{5}(x) \]
Thus, we can rewrite the limit as:
\[ \lim_{x \to 1} \left( 1 + \log_{5}(x) \right)^{\log_{x}(5)} \]

### Step 3: Change the base of the logarithm
Next, we can express \( \log_{x}(5) \) in terms of natural logarithms:
\[ \log_{x}(5) = \frac{\log(5)}{\log(x)} \]
Now our limit becomes:
\[ \lim_{x \to 1} \left( 1 + \log_{5}(x) \right)^{\frac{\log(5)}{\log(x)}} \]

### Step 4: Analyze the limit
As \( x \to 1 \), \( \log_{5}(x) \to 0 \) and \( \log(x) \to 0 \). This means we have an indeterminate form of the type \( 1^{\infty} \). To resolve this, we can take the natural logarithm of the expression:
\[ y = \left( 1 + \log_{5}(x) \right)^{\frac{\log(5)}{\log(x)}} \]
Taking the natural logarithm gives:
\[ \ln(y) = \frac{\log(5)}{\log(x)} \cdot \ln(1 + \log_{5}(x)) \]

### Step 5: Use L'Hôpital's Rule
Now, we need to evaluate:
\[ \lim_{x \to 1} \frac{\log(5) \cdot \ln(1 + \log_{5}(x))}{\log(x)} \]
This is again an indeterminate form \( \frac{0}{0} \). We can apply L'Hôpital's Rule:
1. Differentiate the numerator and denominator with respect to \( x \).
2. The derivative of \( \ln(1 + \log_{5}(x)) \) is \( \frac{1}{1 + \log_{5}(x)} \cdot \frac{1}{x \ln(5)} \).
3. The derivative of \( \log(x) \) is \( \frac{1}{x} \).

### Step 6: Evaluate the limit
After applying L'Hôpital's Rule, we can evaluate the limit:
\[ \lim_{x \to 1} \frac{\log(5) \cdot \frac{1}{1 + \log_{5}(x)} \cdot \frac{1}{x \ln(5)}}{\frac{1}{x}} = \lim_{x \to 1} \frac{\log(5)}{(1 + \log_{5}(x)) \ln(5)} \]
As \( x \to 1 \), \( \log_{5}(x) \to 0 \), so this simplifies to:
\[ \frac{\log(5)}{1 \cdot \ln(5)} = 1 \]

### Step 7: Exponentiate to find \( y \)
Since \( \ln(y) \to 1 \), we have:
\[ y \to e^{1} = e \]

Thus, the final answer is:
\[ \lim_{x \to 1} \left( \log_{5}(5x) \right)^{\log_{x}(5)} = e \]