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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?

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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.

Find the limit, if it exists: \lim_{{x \to \infty}} \frac{90x^6 + 14x + 8}{153x^6 + 3x - 11}. Note: Give the exact answer but not the decimal approximation.

Learn how to calculate the limit of the rational function \lim_{{x \to \infty}} \frac{90x^6 + 14x + 8}{153x^6 + 3x - 11}. This problem involves identifying dominant terms in polynomials and simplifying the expression step by step. Detailed explanation included, suitable for students in grades 11-12.

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