We are asked to determine the following limit:

$\lim_{x \to \infty} \frac{36x^3 + 2x^2 - 4x}{30x^3 + x^2 + 3x + 3}$

Step-by-step solution:

1. Identify the highest degree terms:
   Both the numerator and the denominator contain terms of different powers of $x$, but as $x \to \infty$, the highest degree terms (those involving $x^3$) will dominate.

   • In the numerator, the highest degree term is $36x^3$.
   • In the denominator, the highest degree term is $30x^3$.

2. Simplify the expression:
   We can factor out $x^3$ from both the numerator and denominator:

   $\frac{36x^3 + 2x^2 - 4x}{30x^3 + x^2 + 3x + 3} = \frac{x^3(36 + \frac{2}{x} - \frac{4}{x^2})}{x^3(30 + \frac{1}{x} + \frac{3}{x^2} + \frac{3}{x^3})}$

3. Cancel the common factor of $x^3$: $= \frac{36 + \frac{2}{x} - \frac{4}{x^2}}{30 + \frac{1}{x} + \frac{3}{x^2} + \frac{3}{x^3}}$

4. Take the limit as $x \to \infty$: As $x$ approaches infinity, all terms involving fractions with $x$ in the denominator (like $\frac{2}{x}$, $\frac{4}{x^2}$, etc.) approach zero.

   Thus, we get: $\lim_{x \to \infty} \frac{36 + \frac{2}{x} - \frac{4}{x^2}}{30 + \frac{1}{x} + \frac{3}{x^2} + \frac{3}{x^3}} = \frac{36}{30} = \frac{6}{5}$

Final Answer:

$\lim_{x \to \infty} \frac{36x^3 + 2x^2 - 4x}{30x^3 + x^2 + 3x + 3} = \frac{6}{5}$

Would you like any further explanation or have additional questions about this problem?

Further exploration:

1. How does the behavior of the function change as $x \to -\infty$?
2. What would the limit be if the highest degree term in the numerator and denominator had different powers (e.g., if it was $x^2$)?
3. How do you approach limits involving rational functions when the degrees of the terms in the numerator and denominator are equal?
4. What would happen if the terms involving $x^3$ had opposite signs?
5. How can this limit help us understand asymptotic behavior in other rational functions?

Tip: When working with limits of rational functions, focus on the highest degree terms in both the numerator and denominator, as they will dominate the behavior of the function as $x \to \infty$.