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The limit you provided is:

\[
\lim_{{x \to \infty}} \frac{4 - 11\sqrt{x}}{6 + 3\sqrt{x}}.
\]

### Step-by-step solution:

1. **Factor out** \( \sqrt{x} \) from both the numerator and the denominator.
   \[
   = \lim_{{x \to \infty}} \frac{\frac{4}{\sqrt{x}} - 11}{\frac{6}{\sqrt{x}} + 3}.
   \]
   
2. As \( x \to \infty \), both \( \frac{4}{\sqrt{x}} \) and \( \frac{6}{\sqrt{x}} \) approach 0 because the square root of \( x \) grows large, making these fractions approach 0.

3. Now, the expression simplifies to:
   \[
   = \lim_{{x \to \infty}} \frac{0 - 11}{0 + 3} = \frac{-11}{3}.
   \]

Thus, the value of the limit is:

\[
\boxed{-\frac{11}{3}}.
\]

Would you like more details on any specific step or clarification? Here are some related questions:

1. What happens when the highest-order term dominates in limits?
2. How would you approach limits involving other square roots or higher powers?
3. Why do terms involving \( \frac{1}{\sqrt{x}} \) tend to zero as \( x \to \infty \)?
4. Can this method of factoring out the dominant term work for polynomial functions as well?
5. What are common mistakes when simplifying limits involving square roots?

**Tip**: When dealing with limits at infinity, always check for the dominant term in both the numerator and denominator to simplify the expression effectively.

Find the limit of (4 - 11√x) / (6 + 3√x) as x approaches infinity.

Learn how to solve the limit problem involving radicals as x approaches infinity. This problem covers key concepts of limits, simplifying radical expressions, and the use of dominant terms to find the limit of (4 - 11√x) / (6 + 3√x). Suitable for advanced high school students.

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