Let's go through both limits from the image:

Problem (a):

You are asked to evaluate the limit: $\lim_{x \to \infty} \frac{\sqrt{11 + 9x^2}}{5 + 10x}$

Step 1: Simplify the expression for large $x$

For large $x$, the term $9x^2$ dominates inside the square root. So, we approximate: $\sqrt{11 + 9x^2} \approx \sqrt{9x^2} = 3x \quad \text{as} \quad x \to \infty$ Thus, the expression simplifies to: $\frac{3x}{5 + 10x}$

Step 2: Factor out $x$ from both the numerator and denominator

$\frac{3x}{5 + 10x} = \frac{3x}{x(10 + \frac{5}{x})} = \frac{3}{10 + \frac{5}{x}}$

As $x \to \infty$, $\frac{5}{x} \to 0$. So the limit becomes: $\lim_{x \to \infty} \frac{3}{10} = \frac{3}{10}$

Thus, the limit for part (a) is $\frac{3}{10}$, and the corresponding horizontal asymptote is $y = \frac{3}{10}$.

Problem (b):

You are asked to evaluate the limit: $\lim_{x \to -\infty} \frac{\sqrt{11 + 9x^2}}{5 + 10x}$

Step 1: Simplify the expression for large negative $x$

Again, for large negative $x$, the term $9x^2$ dominates inside the square root. So, approximate: $\sqrt{11 + 9x^2} \approx 3|x| = 3(-x) \quad \text{since} \quad x \to -\infty$ Thus, the expression simplifies to: $\frac{3(-x)}{5 + 10x}$

Step 2: Factor out $x$

$\frac{3(-x)}{5 + 10x} = \frac{-3x}{x(10 + \frac{5}{x})} = \frac{-3}{10 + \frac{5}{x}}$

As $x \to -\infty$, $\frac{5}{x} \to 0$. So the limit becomes: $\lim_{x \to -\infty} \frac{-3}{10} = \frac{-3}{10}$

Thus, the limit for part (b) is $\frac{-3}{10}$, and the corresponding horizontal asymptote is $y = \frac{-3}{10}$.

Would you like more details on any of these steps? Here are five related questions you could explore:

1. How do you find vertical asymptotes in rational functions?
2. Why does $\sqrt{11 + 9x^2} \approx 3|x|$ for large $x$?
3. What is the significance of horizontal asymptotes in real-world applications?
4. How does factoring out the dominant term help when calculating limits?
5. How would these limits change if the expression had a different degree in the numerator or denominator?

Tip: When finding limits as $x$ approaches infinity, always check for dominant terms that grow the fastest in both the numerator and denominator.