We are tasked with finding the limit:

$\lim_{{x \to \infty}} \frac{90x^6 + 14x + 8}{153x^6 + 3x - 11}$

Step-by-step Solution:

1. Identify the degree of the polynomials in the numerator and denominator:

   • The highest power of $x$ in the numerator is $x^6$.
   • The highest power of $x$ in the denominator is also $x^6$.

2. Divide both the numerator and denominator by $x^6$, the highest degree term in both the numerator and denominator:

$\frac{90x^6 + 14x + 8}{153x^6 + 3x - 11} = \frac{90 + \frac{14}{x^5} + \frac{8}{x^6}}{153 + \frac{3}{x^5} - \frac{11}{x^6}}$

3. Take the limit as $x \to \infty$:
   • As $x$ approaches infinity, terms like $\frac{14}{x^5}$, $\frac{8}{x^6}$, $\frac{3}{x^5}$, and $\frac{11}{x^6}$ all approach 0.
   • Therefore, the expression simplifies to:

$\lim_{{x \to \infty}} \frac{90 + 0 + 0}{153 + 0 - 0} = \frac{90}{153}$

4. Simplify the fraction: $\frac{90}{153} = \frac{30}{51} = \frac{10}{17}$

Thus, the limit is:

$\boxed{\frac{10}{17}}$

Would you like further details or clarifications?

Here are 5 questions related to this solution:

1. How do we determine the dominant term in polynomials?
2. What happens to terms with lower powers of $x$ when $x$ approaches infinity?
3. Can we always simplify limits by dividing by the highest power of $x$ in rational expressions?
4. What conditions might make a limit like this one not exist?
5. How would the approach change if the degrees of the numerator and denominator were different?

Tip:

Always simplify the fraction at the end of the limit calculation to its lowest terms for the most precise answer.