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The given problem is asking to find the value of the limit:

\[
\lim_{x \to \infty} \frac{8x^3 - 2x + 7}{x^4 + 4x + 4}
\]

### Step-by-Step Solution:

1. **Analyze the degree of the numerator and denominator:**
   - The numerator is \(8x^3 - 2x + 7\), where the highest degree term is \(8x^3\).
   - The denominator is \(x^4 + 4x + 4\), where the highest degree term is \(x^4\).

2. **Divide both numerator and denominator by the highest degree term in the denominator, which is \(x^4\):**

   \[
   \lim_{x \to \infty} \frac{\frac{8x^3 - 2x + 7}{x^4}}{\frac{x^4 + 4x + 4}{x^4}}
   \]

3. **Simplify each part:**
   - Numerator:
     \[
     \frac{8x^3 - 2x + 7}{x^4} = \frac{8x^3}{x^4} - \frac{2x}{x^4} + \frac{7}{x^4} = \frac{8}{x} - \frac{2}{x^3} + \frac{7}{x^4}
     \]
   - Denominator:
     \[
     \frac{x^4 + 4x + 4}{x^4} = 1 + \frac{4x}{x^4} + \frac{4}{x^4} = 1 + \frac{4}{x^3} + \frac{4}{x^4}
     \]

4. **Take the limit as \(x \to \infty\):**
   - As \(x \to \infty\), all terms with \( \frac{1}{x}, \frac{1}{x^3}, \frac{1}{x^4} \) go to 0.
   - Therefore, the limit simplifies to:
     \[
     \lim_{x \to \infty} \frac{\frac{8}{x} - \frac{2}{x^3} + \frac{7}{x^4}}{1 + \frac{4}{x^3} + \frac{4}{x^4}} = \frac{0}{1} = 0
     \]

### Final Answer:
\[
\lim_{x \to \infty} \frac{8x^3 - 2x + 7}{x^4 + 4x + 4} = 0
\]

Let me know if you would like further clarification or additional details!

Here are 5 related questions:
1. How do you determine the highest degree term in a polynomial?
2. Why do terms involving \( \frac{1}{x^n} \) approach zero as \(x\) approaches infinity?
3. Can this limit be calculated using L'Hopital's Rule? If yes, how?
4. What happens if the degree of the numerator is higher than the denominator in limits?
5. How do you approach limits where both numerator and denominator have the same degree?

**Tip:** When dealing with limits as \(x \to \infty\), always focus on the highest-degree terms in both the numerator and denominator.

Find the limit as x approaches infinity of (8x^3 - 2x + 7) / (x^4 + 4x + 4).

Explore how to find the limit of the rational function (8x^3 - 2x + 7) / (x^4 + 4x + 4) as x approaches infinity. This detailed step-by-step solution demonstrates the application of limits and rational functions concepts, focusing on the Degree Comparison Theorem. Suitable for high school students in Grades 11-12.

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