The limit problem given is:

$\lim_{{x \to \infty}} \frac{9x^3 - 1}{3x^3 + 10}$

To solve this, we’ll analyze the behavior of the function as $x$ approaches infinity by dividing both the numerator and the denominator by the highest power of $x$ in the denominator, which is $x^3$.

Step-by-Step Solution:

1. Rewrite the expression by dividing each term by $x^3$: $\frac{9x^3 - 1}{3x^3 + 10} = \frac{\frac{9x^3}{x^3} - \frac{1}{x^3}}{\frac{3x^3}{x^3} + \frac{10}{x^3}}$

2. Simplify each term: $= \frac{9 - \frac{1}{x^3}}{3 + \frac{10}{x^3}}$

3. As $x \to \infty$, the terms $\frac{1}{x^3}$ and $\frac{10}{x^3}$ approach zero: $= \frac{9 - 0}{3 + 0} = \frac{9}{3} = 3$

Conclusion:

$\lim_{{x \to \infty}} \frac{9x^3 - 1}{3x^3 + 10} = 3$

Would you like further explanation on any steps, or do you have more questions?

Here are five related questions that can deepen your understanding:

1. How would the limit change if the highest power in the denominator was different from the numerator?
2. What if the coefficient of $x^3$ in the denominator was zero?
3. How do limits behave when dealing with polynomials of different degrees?
4. What are some other methods for evaluating limits at infinity?
5. How would this problem change if $x \to -\infty$ instead?

Tip: When evaluating limits at infinity for rational functions, compare the degrees of the polynomials in the numerator and denominator to determine the behavior.