I see that you've uploaded a math problem image. Let's address both questions provided in the image:

Question 9:

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

To solve this limit as $x \to -\infty$, we need to simplify the expression. The highest degree term inside both square roots is $9x^2$, so we can factor out $x^2$ from both square roots:

$\sqrt{9x^2 + 5x + 5} \approx \sqrt{9x^2} = 3|x| = -3x \quad \text{(as } x \to -\infty \text{, } |x| = -x \text{)}$ $\sqrt{9x^2 - 7x - 4} \approx \sqrt{9x^2} = 3|x| = -3x$

Now subtracting the two dominant terms: $\lim_{x \to -\infty} \left( -3x - (-3x) \right) = 0$

Thus, the value of the limit is: $\boxed{0}$

Question 10:

$\lim_{x \to -\infty} \left( \sqrt{x + 5} - \sqrt{2x - 3} \right)$

For very large negative $x$, the square roots dominate. Let's approximate:

$\sqrt{x+5} \approx \sqrt{x}, \quad \sqrt{2x-3} \approx \sqrt{2x}$

Now subtract these dominant terms: $\lim_{x \to -\infty} \left( \sqrt{x} - \sqrt{2x} \right)$ $= \lim_{x \to -\infty} \left( \sqrt{x}(1 - \sqrt{2}) \right)$

As $x \to -\infty$, $\sqrt{x}(1 - \sqrt{2})$ becomes infinitely negative because $1 - \sqrt{2}$ is negative.

Thus, the value of the limit is: $\boxed{-\infty}$

Let me know if you'd like further clarification or a detailed breakdown of these solutions!

Here are 5 related questions for further practice:

1. How do we evaluate limits involving square roots at infinity?
2. What are common techniques for simplifying expressions involving limits?
3. How does factoring out the highest power help in solving limits at infinity?
4. What is the significance of the leading term when finding limits as $x \to \infty$?
5. How do square roots behave differently for positive and negative infinity?

Tip: Always factor out the highest degree term inside square roots to simplify limits at infinity!