We are tasked with solving the following limit as $x \to \infty$:

$\lim_{x \to \infty} \left( 3x - \frac{3x^3 + 4x^2 + x}{x^2 + x + 4} \right)$

Step 1: Simplify the Rational Expression

We focus first on simplifying the rational expression:

$\frac{3x^3 + 4x^2 + x}{x^2 + x + 4}$

As $x \to \infty$, the highest powers of $x$ will dominate the behavior of both the numerator and the denominator.

• Numerator: $3x^3 + 4x^2 + x$ is dominated by $3x^3$ as $x \to \infty$.
• Denominator: $x^2 + x + 4$ is dominated by $x^2$ as $x \to \infty$.

Thus, for large $x$, the expression behaves like:

$\frac{3x^3}{x^2} = 3x$

Step 2: Analyze the Limit Expression

Now substitute this approximation back into the original limit:

$\lim_{x \to \infty} \left( 3x - \frac{3x^3 + 4x^2 + x}{x^2 + x + 4} \right) \approx \lim_{x \to \infty} \left( 3x - 3x \right)$

This simplifies to:

$\lim_{x \to \infty} 0 = 0$

Final Answer:

The value of the limit is:

$\boxed{0}$

Would you like further clarification on any of the steps? Here are some related questions to explore:

1. How do we rigorously justify ignoring lower-order terms in limits at infinity?
2. What would happen if the degrees of the numerator and denominator were different?
3. Can L'Hopital's Rule be applied to this problem?
4. How does this limit change if the terms in the numerator or denominator are altered?
5. What is the importance of dominant terms when evaluating limits involving rational expressions?

Tip: When dealing with limits involving rational expressions at infinity, always compare the highest power terms in the numerator and denominator to simplify your work.