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The given limit is:

\[
\lim_{x \to 0^+} 9 \tan^{-1}(\ln(x))
\]

Let's evaluate this step by step:

### Step 1: Analyze the argument of the inverse tangent
We need to understand the behavior of \(\ln(x)\) as \(x \to 0^+\).

- As \(x \to 0^+\), the natural logarithm \(\ln(x)\) approaches \(-\infty\).
  
### Step 2: Behavior of \(\tan^{-1}(\ln(x))\) as \(x \to 0^+\)
- \(\tan^{-1}(x)\) approaches \(-\frac{\pi}{2}\) as \(x \to -\infty\).
- Therefore, as \(x \to 0^+\), \(\ln(x) \to -\infty\) implies \(\tan^{-1}(\ln(x)) \to -\frac{\pi}{2}\).

### Step 3: Multiply by 9
- Since we have \(9 \tan^{-1}(\ln(x))\), multiply the limit by 9:

\[
\lim_{x \to 0^+} 9 \tan^{-1}(\ln(x)) = 9 \cdot \left(-\frac{\pi}{2}\right) = -\frac{9\pi}{2}
\]

### Final Answer
\[
\lim_{x \to 0^+} 9 \tan^{-1}(\ln(x)) = -\frac{9\pi}{2}
\]

Would you like to see further details, or do you have any other questions?

Here are 5 related questions to explore:

1. What is the behavior of the natural logarithm \(\ln(x)\) near \(x = 0\)?
2. How does \(\tan^{-1}(x)\) behave as \(x \to \infty\) and \(x \to -\infty\)?
3. What are the key properties of inverse trigonometric functions like \(\tan^{-1}(x)\)?
4. How do limits involving logarithmic and trigonometric functions compare to other types of limits?
5. What techniques can be used to evaluate limits that approach \(\infty\) or \(-\infty\)?

**Tip:** When evaluating limits that involve logarithms approaching \(-\infty\), it's helpful to remember the behavior of trigonometric functions like \(\tan^{-1}(x)\), which are bounded by \(\pm \frac{\pi}{2}\).

Evaluate the limit: lim (x -> 0^+) 9 * arctan(ln(x))

Learn how to evaluate the limit lim (x -> 0^+) 9 * arctan(ln(x)) step by step. This solution explores the behavior of the logarithmic and inverse trigonometric functions as x approaches 0 from the positive side, yielding the final result -9π/2.

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