Let's compute the given limit:

$\lim_{x \to \infty} \left( 3x^5 - 3x^4 + 3x^3 - 3x^2 + x - 1 \right)$

Step-by-step Solution:

1. Identify the dominant term: The term with the highest power of $x$ is $3x^5$. As $x \to \infty$, this term will dominate the behavior of the entire expression because it grows faster than the lower-degree terms.

2. Approximate the limit: Since the dominant term is $3x^5$, the behavior of the entire polynomial for large values of $x$ will be controlled by this term. Thus: $\lim_{x \to \infty} \left( 3x^5 - 3x^4 + 3x^3 - 3x^2 + x - 1 \right) \sim \lim_{x \to \infty} 3x^5$ As $x$ grows larger, $3x^5 \to \infty$.

3. Conclusion: Therefore, the limit of the given function as $x \to \infty$ is $\infty$.

Correct Answer:

$\boxed{\infty}$

Do you need more details or have any other questions? Here are a few related questions:

1. What is the general behavior of polynomials as $x \to \infty$?
2. How can we formally use L'Hôpital's Rule to evaluate limits like this one?
3. What are some examples of limits where the dominant term does not determine the limit?
4. How would the limit change if the highest-degree term had a negative coefficient?
5. Can we compute limits at negative infinity for polynomials?

Tip: In polynomial functions, the term with the highest degree of $x$ determines the limit at infinity.