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Let's analyze the given limit problem:

\[
\lim_{x \to 0} \frac{8x^6 + 3x^4 + 18x}{2x^6 + 9x}
\]

### Step-by-step process:

1. **Factor out the lowest power of \(x\)** from both the numerator and denominator:
   - In the numerator, the lowest power is \(x\), so we factor out \(x\):
     \[
     8x^6 + 3x^4 + 18x = x(8x^5 + 3x^3 + 18)
     \]
   - In the denominator, the lowest power is also \(x\):
     \[
     2x^6 + 9x = x(2x^5 + 9)
     \]

2. **Cancel out the common factor \(x\)** from both the numerator and denominator:
   \[
   \frac{x(8x^5 + 3x^3 + 18)}{x(2x^5 + 9)} = \frac{8x^5 + 3x^3 + 18}{2x^5 + 9}
   \]

3. **Evaluate the limit as \(x \to 0\)**:
   - When \(x \to 0\), the higher powers of \(x\) (i.e., \(x^5\), \(x^3\)) tend to zero, so:
     \[
     \lim_{x \to 0} \frac{8x^5 + 3x^3 + 18}{2x^5 + 9} = \frac{18}{9} = 2
     \]

Thus, the value of the limit is:

\[
\boxed{2}
\]

### Conclusion:
The correct answer is **C. 2**, not **D. ∞**.

---

Would you like further details on any step, or have any other questions?

Here are 5 related questions you might find interesting:
1. How do you handle limits where both numerator and denominator approach 0?
2. What is L'Hopital's rule, and when is it applicable?
3. How do you calculate limits involving infinity?
4. Can all limits be solved by factoring and canceling terms?
5. How does the behavior of higher-order terms affect limits?

**Tip**: Always check if higher powers of \(x\) tend to zero when simplifying limits at \(x \to 0\), as this can simplify the evaluation process.

Determine the value of the limit as x approaches 0 for the function (8x^6 + 3x^4 + 18x) / (2x^6 + 9x).

This problem explores the limit of the function (8x^6 + 3x^4 + 18x) / (2x^6 + 9x) as x approaches 0. Through factoring and simplifying, you will find the value of the limit using basic algebraic principles. Suitable for students in Grades 10-12.

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