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The problem involves finding the limit:

\[
\lim_{x \to \infty} \left( \sqrt{5x^2 + 3x + 3} - \sqrt{6x^2 + 3x - 2} \right)
\]

### Steps for solving:

1. **Factor out the highest power of \(x^2\) inside the square roots**:
   - For the first term: \( \sqrt{5x^2 + 3x + 3} = \sqrt{x^2(5 + \frac{3}{x} + \frac{3}{x^2})} = |x| \sqrt{5 + \frac{3}{x} + \frac{3}{x^2}} \)
   - For the second term: \( \sqrt{6x^2 + 3x - 2} = \sqrt{x^2(6 + \frac{3}{x} - \frac{2}{x^2})} = |x| \sqrt{6 + \frac{3}{x} - \frac{2}{x^2}} \)

   As \(x \to \infty\), we can treat \(|x| = x\), since we are considering positive infinity.

2. **Simplify for large \(x\)**:
   As \(x \to \infty\), the terms \(\frac{3}{x}\), \(\frac{3}{x^2}\), and \(\frac{-2}{x^2}\) approach 0. So the expressions inside the square roots simplify to:

   \[
   \sqrt{5 + \frac{3}{x} + \frac{3}{x^2}} \to \sqrt{5}, \quad \sqrt{6 + \frac{3}{x} - \frac{2}{x^2}} \to \sqrt{6}
   \]

3. **Take the limit**:
   The expression now becomes:

   \[
   \lim_{x \to \infty} \left( x\sqrt{5} - x\sqrt{6} \right)
   \]

   Factor out \(x\):

   \[
   x\left( \sqrt{5} - \sqrt{6} \right)
   \]

   Since \(\sqrt{5} - \sqrt{6}\) is negative, the whole expression tends to \(-\infty\) as \(x \to \infty\).

### Conclusion:
The limit is \( -\infty \), which corresponds to option **c.**

Would you like a detailed explanation of any specific step, or further assistance?

Here are some related questions you might find useful:
1. What happens when the coefficients inside the square roots are equal?
2. How does factoring the highest degree of \(x\) help in limit problems?
3. What are some techniques for handling limits involving square roots at infinity?
4. How does the behavior of functions change as \(x \to -\infty\)?
5. What are common mistakes to avoid when solving limits at infinity?

**Tip:** When solving limits at infinity involving square roots, factoring out the highest power of \(x^2\) often simplifies the process.

lim x→∞ √(5x^2 + 3x + 3) - √(6x^2 + 3x - 2)

Calculate the limit as x approaches infinity for the expression √(5x² + 3x + 3) - √(6x² + 3x - 2). This problem involves using limits, simplifying square roots, and analyzing the asymptotic behavior of functions at infinity.

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