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We are tasked with solving the limit of the given expression as \( x \to \infty \):

\[
\lim_{x \to \infty} \frac{7x^9 - 2x^5 + 9}{2x^3 + 5x^2 - 7}
\]

### Steps to solve:

1. **Determine the dominant term in both the numerator and the denominator**: 
   - In the **numerator**, the highest degree term is \( 7x^9 \).
   - In the **denominator**, the highest degree term is \( 2x^3 \).

2. **Divide every term in both the numerator and the denominator by \( x^9 \)** (since \( x^9 \) is the highest degree of the polynomial in the numerator):

\[
\lim_{x \to \infty} \frac{\frac{7x^9}{x^9} - \frac{2x^5}{x^9} + \frac{9}{x^9}}{\frac{2x^3}{x^9} + \frac{5x^2}{x^9} - \frac{7}{x^9}}
= \lim_{x \to \infty} \frac{7 - \frac{2}{x^4} + \frac{9}{x^9}}{\frac{2}{x^6} + \frac{5}{x^7} - \frac{7}{x^9}}
\]

3. **Analyze the behavior as \( x \to \infty \)**:
   - Terms like \( \frac{2}{x^4} \), \( \frac{9}{x^9} \), \( \frac{2}{x^6} \), etc., approach 0 as \( x \to \infty \).

4. **Simplify the limit**:
   - The numerator simplifies to \( 7 \) as all other terms go to 0.
   - The denominator simplifies to 0 because all terms go to 0.

So, the expression becomes:

\[
\lim_{x \to \infty} \frac{7}{0}
\]

Since dividing by zero leads to an undefined or infinite result, the limit is:

\[
\boxed{\infty}
\]

### Conclusion:
The limit of the given expression as \( x \to \infty \) is \( \infty \).

Would you like more details on any step or further clarifications?

### Related questions:
1. What would happen if the highest degree term in the denominator were higher than in the numerator?
2. How does the behavior of limits change if we were looking at \( x \to -\infty \)?
3. Can L'Hopital's Rule be applied to this limit?
4. What are the steps to analyze a rational function's behavior at \( x = \infty \)?
5. How do asymptotes relate to limits at infinity in rational functions?

### Tip:
Always check the degrees of the polynomial terms in limits involving rational functions, as they usually determine the behavior at infinity.

Find the limit as x approaches infinity for the expression: (7x^9 - 2x^5 + 9) / (2x^3 + 5x^2 - 7).

Learn how to solve the limit of the rational function (7x^9 - 2x^5 + 9) / (2x^3 + 5x^2 - 7) as x approaches infinity. This problem involves understanding polynomial degree dominance and simplifying limits using the highest degree terms. Suitable for students in Grades 10-12.

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