limb→5−−ln(|b|)25+1/5b）+ln|b−5|25−1/5−ln(4)25

The given expression involves limits and logarithmic functions. It appears to be somewhat complex, but I will break it down and clarify the expression first before evaluating the limit.

The expression provided seems to be:

\[
\lim_{b \to 5} \left( - \frac{\ln(|b|)}{25 + \frac{1}{5b}} + \frac{\ln|b-5|}{25 - \frac{1}{5}} - \frac{\ln(4)}{25} \right)
\]

### Steps to Simplify and Solve:
1. **Break the expression into parts:**
   - First term: \(\frac{\ln(|b|)}{25 + \frac{1}{5b}}\)
   - Second term: \(\frac{\ln|b - 5|}{25 - \frac{1}{5}}\)
   - Third term: \(\frac{\ln(4)}{25}\)

2. **Evaluate each term as \(b \to 5\):**

   - **First term:** 
     \[
     \lim_{b \to 5} \frac{\ln(|b|)}{25 + \frac{1}{5b}} = \frac{\ln(5)}{25 + \frac{1}{25}} = \frac{\ln(5)}{25.04}
     \]

   - **Second term:**
     \[
     \lim_{b \to 5} \frac{\ln|b - 5|}{25 - \frac{1}{5}} \quad \text{as} \quad b \to 5, \, |b - 5| \to 0
     \]
     The logarithmic term \(\ln|b - 5|\) approaches \(-\infty\), so this part tends towards 0 due to division by the constant denominator.
     
   - **Third term:**
     This is constant and equals \(\frac{\ln(4)}{25}\).

3. **Final Expression:**

   The limit reduces to:
   \[
   - \frac{\ln(5)}{25.04} + 0 - \frac{\ln(4)}{25}
   \]

4. **Simplifying further:**

   Combine the remaining terms:
   \[
   = - \left(\frac{\ln(5)}{25.04} + \frac{\ln(4)}{25}\right)
   \]

This is the simplified form of the limit.

Would you like further details on any specific part or have any questions?

### Related Questions:
1. What are the properties of logarithmic functions in limits?
2. How do we handle indeterminate forms like \(\frac{0}{0}\) in limits involving logarithms?
3. What are the steps for evaluating limits involving absolute values and logarithms?
4. How do we deal with expressions that tend towards infinity in limits?
5. What is the importance of simplifying expressions before evaluating limits?

#### Tip:
Always break down complex limits into manageable parts and evaluate them individually before combining the results.

lim→5−−ln(|b|)25+1/5b）+ln|b−5|25−1/5−ln(4)25

Evaluate the limit of an expression involving logarithmic functions and absolute values as b approaches 5. This step-by-step solution breaks down the complex expression into manageable parts, utilizing properties of logarithms and limits. Suitable for advanced high school students (Grades 11-12).

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