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The image shows the following limit problem:

\[
\lim_{{x \to \infty}} \frac{3x^2 - 6x + 1}{2 - 5x - 6x^2}
\]

### Step-by-step solution:

1. **Identify the highest degree of \(x\) in both the numerator and denominator:**
   - The highest power in the numerator is \(x^2\) from \(3x^2\).
   - The highest power in the denominator is also \(x^2\) from \(-6x^2\).

2. **Factor out \(x^2\) from both the numerator and the denominator:**
   - Numerator: \(3x^2 - 6x + 1 = x^2(3 - \frac{6}{x} + \frac{1}{x^2})\)
   - Denominator: \(2 - 5x - 6x^2 = x^2(-6 - \frac{5}{x} + \frac{2}{x^2})\)

3. **Simplify the limit expression:**
   \[
   \lim_{{x \to \infty}} \frac{x^2(3 - \frac{6}{x} + \frac{1}{x^2})}{x^2(-6 - \frac{5}{x} + \frac{2}{x^2})}
   \]
   - Cancel \(x^2\) from both numerator and denominator:
   \[
   = \lim_{{x \to \infty}} \frac{3 - \frac{6}{x} + \frac{1}{x^2}}{-6 - \frac{5}{x} + \frac{2}{x^2}}
   \]

4. **Evaluate the limit as \(x \to \infty\):**
   - As \(x \to \infty\), the terms \(\frac{6}{x}\), \(\frac{1}{x^2}\), \(\frac{5}{x}\), and \(\frac{2}{x^2}\) approach 0.
   - Therefore, the expression becomes:
   \[
   \frac{3 - 0 + 0}{-6 - 0 + 0} = \frac{3}{-6} = -\frac{1}{2}
   \]

### Final Answer:
\[
\lim_{{x \to \infty}} \frac{3x^2 - 6x + 1}{2 - 5x - 6x^2} = -\frac{1}{2}
\]

Would you like more details on any step? Here are 5 related questions to deepen your understanding:

1. What happens if the highest degree terms in the numerator and denominator are different?
2. How would the limit change if \(x \to -\infty\)?
3. Can L'Hopital's rule be used to solve this limit problem?
4. How do limits at infinity behave in rational functions?
5. What does it mean geometrically when the limit of a rational function tends to a constant?

**Tip:** When solving limits at infinity, focus on the highest-degree terms in both the numerator and denominator, as lower-degree terms tend to zero.

Find the limit of (3x^2 - 6x + 1) / (2 - 5x - 6x^2) as x approaches infinity.

This solution explains how to find the limit of the rational function (3x^2 - 6x + 1) / (2 - 5x - 6x^2) as x approaches infinity. The problem focuses on factoring out the highest powers of x and simplifying, using limit theorems. Suitable for students at a precalculus or early calculus level.

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