What is `lim_(xto0) (log_(5)(1+x))/(x)` equal to?

Options

A. 1
B. `log_(5)e`
C. `log_(e)5`
D. 5

Explanation

To solve the limit \( \lim_{x \to 0} \frac{\log_{5}(1+x)}{x} \), we can follow these steps:

### Step 1: Change of Base Formula
We can use the change of base formula for logarithms, which states that:
\[ \log_{b}(a) = \frac{\log_{e}(a)}{\log_{e}(b)} \]
Applying this to our limit, we rewrite \( \log_{5}(1+x) \):
\[ \log_{5}(1+x) = \frac{\log_{e}(1+x)}{\log_{e}(5)} \]

### Step 2: Substitute into the Limit
Now substitute this back into the limit:
\[ \lim_{x \to 0} \frac{\log_{5}(1+x)}{x} = \lim_{x \to 0} \frac{\frac{\log_{e}(1+x)}{\log_{e}(5)}}{x} = \frac{1}{\log_{e}(5)} \lim_{x \to 0} \frac{\log_{e}(1+x)}{x} \]

### Step 3: Evaluate the Limit
We know from calculus that:
\[ \lim_{x \to 0} \frac{\log_{e}(1+x)}{x} = 1 \]
Thus, we can substitute this result into our expression:
\[ \frac{1}{\log_{e}(5)} \cdot 1 = \frac{1}{\log_{e}(5)} \]

### Step 4: Change Back to Logarithm Base 5
Using the property of logarithms, we can express \( \frac{1}{\log_{e}(5)} \) as:
\[ \frac{1}{\log_{e}(5)} = \log_{5}(e) \]

### Final Answer
Thus, the limit is:
\[ \lim_{x \to 0} \frac{\log_{5}(1+x)}{x} = \log_{5}(e) \]