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The given expression asks to evaluate the limit as \( x \to \infty \) of the following function:

\[
\lim_{x \to \infty} \sqrt{\frac{(3x^3 + 5x - 1)^3 (9 - 2x^2)^2}{(2x + 7) (4x^4 - 5)^3}}
\]

### Step-by-step breakdown:

1. **Analyze the asymptotic behavior**:
   - As \( x \to \infty \), we focus on the highest degree terms in both the numerator and denominator, since those will dominate the behavior of the expression.
   
   - **Numerator**:
     - \( 3x^3 + 5x - 1 \sim 3x^3 \)
     - \( 9 - 2x^2 \sim -2x^2 \)
     Therefore, the numerator behaves like:
     \[
     (3x^3)^3 (-2x^2)^2 = 27x^9 \cdot 4x^4 = 108x^{13}
     \]

   - **Denominator**:
     - \( 2x + 7 \sim 2x \)
     - \( 4x^4 - 5 \sim 4x^4 \)
     Therefore, the denominator behaves like:
     \[
     2x \cdot (4x^4)^3 = 2x \cdot 64x^{12} = 128x^{13}
     \]

2. **Simplify the asymptotic limit**:
   So, the expression inside the square root becomes asymptotically:
   \[
   \frac{108x^{13}}{128x^{13}} = \frac{108}{128} = \frac{27}{32}
   \]

3. **Take the square root**:
   \[
   \sqrt{\frac{27}{32}} = \frac{\sqrt{27}}{\sqrt{32}} = \frac{3\sqrt{3}}{4\sqrt{2}} = \frac{3\sqrt{6}}{8}
   \]

Thus, the value of the limit is:

\[
\boxed{\frac{3\sqrt{6}}{8}}
\]

Let me know if you would like further details on any of the steps!

### Related questions:
1. How would the solution change if the degree of the polynomial in the numerator was higher?
2. What happens to the limit if the denominator had an additional higher-degree term?
3. How can you determine the dominant term for complex functions as \( x \to \infty \)?
4. Can L'Hopital's Rule be applied to this type of limit problem?
5. What is the behavior of this function as \( x \to -\infty \)?

### Tip:
When evaluating limits at infinity, always focus on the highest power terms, as they dominate the behavior of the function.

Evaluate lim (x → ∞) √((3x^3 + 5x - 1)^3 (9 - 2x^2)^2 / ((2x + 7)(4x^4 - 5)^3))

Learn how to evaluate the limit of a complex polynomial ratio as x approaches infinity, focusing on the dominant terms. This example demonstrates key concepts such as asymptotic behavior, polynomial division, and square root simplifications in advanced calculus.

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