https://file.aiquickdraw.com/mathbot/file/1727310486947q2udy17a.jpg

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.

Evaluate the limits of the following expressions involving square roots and rational functions, and find the corresponding horizontal asymptotes.

Learn how to evaluate limits involving square roots and rational functions as x approaches infinity. This solution also covers the method to find horizontal asymptotes for such expressions. Explore key concepts in limits, asymptotes, and dominant term approximation.

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