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Let's solve the given limit:

\[
\lim_{x \to +\infty} \sqrt{x^3} \left[ \frac{3}{\sqrt{x}} - \arctan \left( \frac{3}{\sqrt{x}} \right) \right]
\]

### Step-by-step Solution:

1. **Simplify the expression inside the limit:**
   - We can rewrite \(\sqrt{x^3}\) as \(x^{3/2}\).
   - Thus, the expression becomes:
   \[
   \lim_{x \to +\infty} x^{3/2} \left[ \frac{3}{\sqrt{x}} - \arctan \left( \frac{3}{\sqrt{x}} \right) \right]
   \]

2. **Simplify \(\frac{3}{\sqrt{x}}\):**
   - \(\frac{3}{\sqrt{x}} = 3x^{-1/2}\), so now the limit becomes:
   \[
   \lim_{x \to +\infty} x^{3/2} \left[ 3x^{-1/2} - \arctan \left( 3x^{-1/2} \right) \right]
   \]

3. **Evaluate the limit as \(x \to +\infty\):**
   - As \(x \to +\infty\), \(x^{-1/2} \to 0\).
   - Hence, \(3x^{-1/2} \to 0\), and \(\arctan(3x^{-1/2}) \to 0\) since \(\arctan(y) \to 0\) as \(y \to 0\).
   - Thus, the expression inside the square brackets approaches:
     \[
     3x^{-1/2} - \arctan(3x^{-1/2}) \to 0 \text{ as } x \to +\infty.
     \]
   - Now let's handle the outer factor \(x^{3/2}\).

4. **Analyze the asymptotic behavior:**
   - For large \(x\), the difference \(3x^{-1/2} - \arctan(3x^{-1/2})\) behaves like \(3x^{-1/2}\), since \(\arctan(z) \approx z\) for small \(z\).
   - Hence, for large \(x\), the product \(x^{3/2} \cdot 3x^{-1/2}\) behaves like \(3x\).

5. **Final limit:**
   - Therefore, the limit as \(x \to +\infty\) is:
   \[
   \lim_{x \to +\infty} x^{3/2} \cdot 3x^{-1/2} = 3x.
   \]
   - As \(x \to +\infty\), this grows without bound. Therefore, the limit is:

\[
\boxed{\infty}.
\]

Would you like more details or clarifications?

### Related Questions:
1. How does \(\arctan(x)\) behave for very large and very small values of \(x\)?
2. What happens if we change the power of \(x\) in \(\sqrt{x^3}\) to another power?
3. How would the limit change if we used a different trigonometric function like \(\sin^{-1}(x)\)?
4. How do we apply L'Hopital's Rule in similar limit problems involving infinity?
5. What are the key asymptotic behaviors of functions involving \(\arctan(x)\) and \(x^{n/2}\)?

### Tip:
When dealing with limits at infinity involving complex functions like \(\arctan\), approximations for small or large values can simplify the problem dramatically.

lim (x -> +∞) √x^3 [ (3 / √x) - arctan (3 / √x) ]

Explore the step-by-step solution to the limit problem involving √x^3, trigonometric arctan, and asymptotic behavior as x approaches infinity. This problem is suitable for undergraduate students familiar with advanced calculus and trigonometric approximations.

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