The value of `lim_(xrarr1)(log_5 5x)^(log_x5)` , is

Options

A. 1
B. e
C. -1
D. none of these

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 Limit
We start by rewriting the limit:
\[ \lim_{x \to 1} \left( \log_5 (5x) \right)^{\log_x 5} \]

### Step 2: Use Logarithmic Properties
Using the change of base formula for logarithms, we can express \( \log_x 5 \) as:
\[ \log_x 5 = \frac{\log_e 5}{\log_e x} \]
Thus, we can rewrite the limit as:
\[ \lim_{x \to 1} \left( \log_5 (5x) \right)^{\frac{\log_e 5}{\log_e x}} \]

### Step 3: Simplify \( \log_5 (5x) \)
Now, we simplify \( \log_5 (5x) \):
\[ \log_5 (5x) = \log_5 5 + \log_5 x = 1 + \log_5 x \]
So, we can substitute this back into our limit:
\[ \lim_{x \to 1} \left( 1 + \log_5 x \right)^{\frac{\log_e 5}{\log_e x}} \]

### Step 4: Identify the Indeterminate Form
As \( x \to 1 \), \( \log_5 x \to 0 \) and \( \log_e x \to 0 \). Therefore, we have the form \( 1^{\infty} \), which is an indeterminate form.

### Step 5: Convert to Exponential Form
To resolve this indeterminate form, we can use the fact that:
\[ a^{b} = e^{b \ln a} \]
Thus, we rewrite our limit as:
\[ \lim_{x \to 1} e^{\frac{\log_e 5}{\log_e x} \ln(1 + \log_5 x)} \]

### Step 6: Simplify the Exponent
Now we need to find:
\[ \lim_{x \to 1} \frac{\log_e 5}{\log_e x} \ln(1 + \log_5 x) \]
Using the approximation \( \ln(1 + u) \approx u \) for small \( u \):
\[ \ln(1 + \log_5 x) \approx \log_5 x \]
Thus, we can rewrite the limit as:
\[ \lim_{x \to 1} \frac{\log_e 5 \cdot \log_5 x}{\log_e x} \]

### Step 7: Change of Base for \( \log_5 x \)
Using the change of base formula again:
\[ \log_5 x = \frac{\log_e x}{\log_e 5} \]
Substituting this into our limit gives:
\[ \lim_{x \to 1} \frac{\log_e 5 \cdot \frac{\log_e x}{\log_e 5}}{\log_e x} = \lim_{x \to 1} 1 = 1 \]

### Step 8: Final Result
Thus, we have:
\[ \lim_{x \to 1} e^{1} = e \]

### Conclusion
The value of the limit is:
\[ \boxed{e} \]