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The mathematical problem you've shared is a limit problem involving a complex algebraic expression:

\[ \lim_{x \to \infty} \frac{4x^2 - 2024ax + 4048a^2 (12x^2 + 2x + 2024y)}{\sqrt{16x^4 - 4048 \sqrt{x^3 - 2096x^2 - 3033}} } = p \]

To find \( p \), and then use that to determine \( 3p - 1 \), we can simplify and analyze the behavior of the expression as \( x \) approaches infinity. Here's the breakdown:

1. **Numerator Simplification**:
   - Simplify the terms in the numerator and examine the leading coefficients and powers of \( x \) since higher powers dominate as \( x \) approaches infinity.

2. **Denominator Simplification**:
   - The denominator is more complex, involving nested square roots and powers of \( x \). Simplify the inner square root and the outer square root expressions, again focusing on the dominant terms as \( x \) becomes very large.

Let's start by simplifying both parts of the expression:

### Step 1: Simplify the Numerator
The numerator:
\[ 4x^2 - 2024ax + 4048a^2(12x^2 + 2x + 2024y) \]
Expanding and simplifying:
\[ 4x^2 - 2024ax + 48576a^2x^2 + 8096a^2x + 8189696a^2y \]
Combining like terms:
\[ (4x^2 + 48576a^2x^2) - 2024ax + 8096a^2x + 8189696a^2y \]

### Step 2: Simplify the Denominator
The denominator:
\[ \sqrt{16x^4 - 4048 \sqrt{x^3 - 2096x^2 - 3033}} \]
Focus on the highest power terms as \( x \to \infty \):
The inner square root:
\[ \sqrt{x^3 - 2096x^2 - 3033} \approx \sqrt{x^3} = x^{3/2} \]
The entire expression then simplifies to:
\[ \sqrt{16x^4 - 4048x^{3/2}} \approx \sqrt{16x^4} = 4x^2 \]

### Step 3: Evaluate the Limit
Combining the simplified numerator and denominator:
\[ \lim_{x \to \infty} \frac{48580a^2x^2 - 2024ax + 8096a^2x + 8189696a^2y}{4x^2} \]

The leading term in the numerator is \( 48580a^2x^2 \) and in the denominator is \( 4x^2 \), thus:
\[ \lim_{x \to \infty} \frac{48580a^2x^2}{4x^2} = 12145a^2 \]

Assuming \( p = 12145a^2 \), then:
\[ 3p - 1 = 3(12145a^2) - 1 = 36435a^2 - 1 \]

Would you like further details on any specific part of this calculation, or do you have other questions?

**Further questions you might consider:**
1. What happens if the coefficients of \( x \) and \( a \) are changed in the original problem?
2. How does the addition of \( y \) affect the limit as \( x \) approaches infinity?
3. Could the presence of square roots affect the continuity or differentiability of this expression at certain points?
4. What are the implications of this type of limit in practical applications?
5. How would this problem change if it were set to the limit as \( x \) approaches zero instead of infinity?

**Tip:** When dealing with limits that include both polynomials and radical expressions, focus on the highest degree terms, as these will dictate the behavior of the function as \( x \) approaches large values or infinity.

Jika lim x→∞ (4x^2 − 2024ax + 4048a^2)(12x^2 + 2x + 2024y) ÷ (√(16x^4 − 4048 − √(x^3 − 2096x^2 − 3033))^2) = p, maka nilai dari 3p - 1 = ...

This page provides a step-by-step solution to a limit problem involving polynomials and nested radical expressions. Learn how to simplify complex algebraic expressions, apply the limit theorem, and understand the behavior of functions as x approaches infinity. Suitable for high school students in Grades 11-12.

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