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To calculate the limit of the sequence given by

\[
x_n = \frac{n^2 + 3n}{n - 9} + \frac{4n^2 - n^3}{n^2 - 81},
\]

we can analyze each term separately as \( n \to \infty \).

### Step-by-Step Solution

1. **First Term: \( \frac{n^2 + 3n}{n - 9} \)**
   - As \( n \to \infty \), the higher power of \( n \) in the numerator and denominator dominates.
   - So, we simplify by dividing both the numerator and denominator by \( n \):
     \[
     \frac{n^2 + 3n}{n - 9} = \frac{n^2 \left(1 + \frac{3}{n}\right)}{n \left(1 - \frac{9}{n}\right)} = \frac{n \cdot (1 + \frac{3}{n})}{1 - \frac{9}{n}}.
     \]
   - As \( n \to \infty \), terms with \( \frac{1}{n} \) tend to zero, leaving:
     \[
     \frac{n \cdot 1}{1} = n.
     \]

2. **Second Term: \( \frac{4n^2 - n^3}{n^2 - 81} \)**
   - Here, \( n^3 \) in the numerator and \( n^2 \) in the denominator are the dominant terms as \( n \to \infty \).
   - Simplify by dividing the numerator and denominator by \( n^2 \):
     \[
     \frac{4n^2 - n^3}{n^2 - 81} = \frac{n^2 \left(4 - \frac{n}{n^2}\right)}{n^2 \left(1 - \frac{81}{n^2}\right)} = \frac{4 - n}{1 - 0} = -n.
     \]

3. **Combine Results**
   - The first term approaches \( n \) and the second term approaches \( -n \).
   - Therefore, as \( n \to \infty \):
     \[
     x_n = n + (-n) = 0.
     \]

### Final Answer
\[
\lim_{{n \to \infty}} x_n = 0.
\]

Would you like further details or have questions on any step?

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Here are related questions to expand your understanding:

1. What happens if we modify the power of \( n \) in the numerator of each fraction?
2. How do we evaluate limits with terms that grow at different rates?
3. Can the result change if we add a constant term to each fraction?
4. What other techniques could simplify such limits?
5. How does the limit change if we replace \( n \) with \( n^2 \) in certain terms?

**Tip:** When finding limits at infinity, focus on the highest power terms, as they dominate the behavior of the function.

Calculate the limit of the sequence x_n = (n^2 + 3n) / (n - 9) + (4n^2 - n^3) / (n^2 - 81).

Solve a sequence limit problem involving rational expressions as n approaches infinity. This solution covers the dominant term analysis technique and limit comparison for sequences. Suitable for undergraduate calculus students.

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