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To evaluate this limit as \( x \to \infty \):

\[
\lim_{x \to \infty} \frac{(3x^4 + 1)(x^2 + 3)}{5x^5 + 7x^2 + 8}
\]

we'll analyze the degrees of the polynomial in the numerator and the denominator.

### Step-by-Step Solution

1. **Expand the numerator**: The numerator is \( (3x^4 + 1)(x^2 + 3) \). Let's expand it:
   \[
   (3x^4 + 1)(x^2 + 3) = 3x^6 + 9x^4 + x^2 + 3
   \]

   So, the numerator becomes:
   \[
   3x^6 + 9x^4 + x^2 + 3
   \]

2. **Identify the highest degree term in the numerator**: The highest degree term in the expanded numerator is \( 3x^6 \).

3. **Identify the highest degree term in the denominator**: The denominator is \( 5x^5 + 7x^2 + 8 \), where the highest degree term is \( 5x^5 \).

4. **Simplify the limit by dividing by the highest power of \( x \) in the denominator**:
   \[
   \lim_{x \to \infty} \frac{3x^6 + 9x^4 + x^2 + 3}{5x^5 + 7x^2 + 8}
   \]

   Divide both the numerator and the denominator by \( x^5 \):
   \[
   = \lim_{x \to \infty} \frac{3x + \frac{9}{x} + \frac{1}{x^3} + \frac{3}{x^5}}{5 + \frac{7}{x^3} + \frac{8}{x^5}}
   \]

5. **Evaluate the limit**: As \( x \to \infty \), all terms with \( \frac{1}{x^k} \) (where \( k > 0 \)) will approach zero. This leaves:
   \[
   = \frac{3x}{5} \to \infty
   \]

### Conclusion
The limit diverges to infinity:
\[
\lim_{x \to \infty} \frac{(3x^4 + 1)(x^2 + 3)}{5x^5 + 7x^2 + 8} = \infty
\]

Would you like a deeper explanation or have any questions?

---

Here are five related questions to consider:

1. How do we determine the behavior of polynomials as \( x \to \infty \)?
2. Why do terms with lower powers of \( x \) become negligible in limits as \( x \to \infty \)?
3. What are some strategies to find limits at infinity for rational functions?
4. How would the result change if the highest degree of \( x \) in the denominator was higher than in the numerator?
5. How does L'Hôpital's Rule apply in limits at infinity when both numerator and denominator approach infinity?

**Tip:** For rational functions, focusing on the highest power of \( x \) in the numerator and denominator can simplify limits as \( x \to \infty \).

lim (x -> infinity) ((3x^4 + 1)(x^2 + 3)) / (5x^5 + 7x^2 + 8)

Learn how to evaluate the limit at infinity for the rational function ((3x^4 + 1)(x^2 + 3)) / (5x^5 + 7x^2 + 8). This solution provides a step-by-step explanation on analyzing polynomial degrees to determine the limit behavior. Suitable for advanced high school students in Grades 11-12.

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