The problem involves evaluating the limit:

$\lim_{x \to \infty} \left( \sqrt{9x^2 + 5x + 5} - \sqrt{9x^2 - 7x - 4} \right).$

Solution:

1. Factor out $x^2$ from each square root: For large $x$, we can simplify the square roots by factoring $x^2$ out from each term under the root:

   $\sqrt{9x^2 + 5x + 5} = \sqrt{x^2 \left( 9 + \frac{5}{x} + \frac{5}{x^2} \right)} = x \sqrt{9 + \frac{5}{x} + \frac{5}{x^2}},$ $\sqrt{9x^2 - 7x - 4} = \sqrt{x^2 \left( 9 - \frac{7}{x} - \frac{4}{x^2} \right)} = x \sqrt{9 - \frac{7}{x} - \frac{4}{x^2}}.$

   Thus, the expression becomes: $\lim_{x \to \infty} \left( x \sqrt{9 + \frac{5}{x} + \frac{5}{x^2}} - x \sqrt{9 - \frac{7}{x} - \frac{4}{x^2}} \right).$

2. Factor $x$ out: Factor $x$ from the entire expression: $x \left( \sqrt{9 + \frac{5}{x} + \frac{5}{x^2}} - \sqrt{9 - \frac{7}{x} - \frac{4}{x^2}} \right).$

3. Use the binomial approximation: For large $x$, the term inside each square root can be expanded using the binomial approximation: $\sqrt{9 + \frac{a}{x} + \frac{b}{x^2}} \approx 3 + \frac{1}{6} \cdot \frac{a}{x}, \quad \text{for small terms involving } \frac{1}{x}.$ Apply this to both terms: $\sqrt{9 + \frac{5}{x} + \frac{5}{x^2}} \approx 3 + \frac{1}{6} \cdot \frac{5}{x},$ $\sqrt{9 - \frac{7}{x} - \frac{4}{x^2}} \approx 3 - \frac{1}{6} \cdot \frac{7}{x}.$

   Substitute these approximations back into the expression: $x \left( \sqrt{9 + \frac{5}{x} + \frac{5}{x^2}} - \sqrt{9 - \frac{7}{x} - \frac{4}{x^2}} \right) \approx x \left( \left( 3 + \frac{5}{6x} \right) - \left( 3 - \frac{7}{6x} \right) \right).$

4. Simplify the result: Simplify the difference inside the parentheses: $\left( 3 + \frac{5}{6x} \right) - \left( 3 - \frac{7}{6x} \right) = \frac{5}{6x} + \frac{7}{6x} = \frac{12}{6x} = \frac{2}{x}.$

   The overall expression becomes: $x \cdot \frac{2}{x} = 2.$

Final Answer:

$\boxed{2}$

Let me know if you'd like further clarification! 😊

Related Questions:

1. How does the binomial expansion simplify square roots?
2. What happens if $x \to -\infty$ in this problem?
3. How do you compute limits involving other square root differences?
4. What are other techniques for handling limits with radicals?
5. Can we use L'Hôpital's Rule here? Why or why not?

Tip:

Always try factoring out the dominant term (e.g., $x^2$) to simplify limits involving radicals for large $x$.