2 lim

x→9+

x − 9

5 + log1/5

(x − 9) =

A −∞

B 0

+

C 5

D +∞

E 0

−

We are tasked with evaluating the limit:

\[
\lim_{x \to 9^+} \left( \frac{x - 9}{5 + \log_{\frac{1}{5}}(x - 9)} \right)
\]

### Step 1: Behavior of \( x - 9 \) as \( x \to 9^+ \)
As \( x \to 9^+ \), the expression \( x - 9 \) becomes a small positive number approaching \( 0^+ \).

### Step 2: Behavior of \( \log_{\frac{1}{5}}(x - 9) \) as \( x \to 9^+ \)
The logarithmic expression \( \log_{\frac{1}{5}}(x - 9) \) involves a logarithm with a base \( \frac{1}{5} \). Recall that logarithms with bases less than 1 have the following behavior:

\[
\log_{\frac{1}{5}}(x - 9) = \frac{\log(x - 9)}{\log(\frac{1}{5})}
\]

Since \( \log(\frac{1}{5}) \) is negative and \( \log(x - 9) \) becomes negative as \( x - 9 \to 0^+ \), the entire logarithmic expression tends to \( +\infty \) as \( x - 9 \to 0^+ \).

### Step 3: Limit of the full expression
Now, we examine the full expression:

\[
\frac{x - 9}{5 + \log_{\frac{1}{5}}(x - 9)}
\]

As \( x - 9 \to 0^+ \):
- The numerator \( x - 9 \to 0^+ \).
- The denominator \( 5 + \log_{\frac{1}{5}}(x - 9) \to +\infty \).

Since the numerator approaches \( 0^+ \) and the denominator approaches \( +\infty \), the overall expression tends to \( 0 \).

### Conclusion:
The limit is:

\[
\lim_{x \to 9^+} \left( \frac{x - 9}{5 + \log_{\frac{1}{5}}(x - 9)} \right) = 0
\]

From the given options, the correct answer is:

\[
\boxed{E \, (0^-)}
\]

This is because the expression approaches \( 0 \) from the negative side (due to the logarithmic behavior).

---

Would you like further details on any part of this explanation?

Here are 5 questions to extend this topic:
1. How do logarithms behave for bases greater than 1 versus less than 1?
2. Can you provide other examples of limits involving logarithmic functions?
3. What are the key differences between one-sided and two-sided limits?
4. How would this limit change if the base of the logarithm were greater than 1?
5. Can you apply L'Hopital's Rule to verify the result?

**Tip:** For limits involving logarithmic functions, always analyze the behavior of the logarithm based on its base and the value inside the log.

2 lim x→9+ x − 9 / 5 + log1/5(x − 9) = A −∞, B 0+, C 5, D +∞, E 0−

This page provides a detailed solution for evaluating the limit as x approaches 9 from the positive side for the expression (x − 9) / (5 + log_1/5(x − 9)). The solution includes the behavior of logarithmic functions with bases less than 1 and their asymptotic properties. Suitable for advanced calculus students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation