It seems you're asking for the limit of the following expression:

\[
\lim_{x \to \infty} \left( x^{15} + \frac{1}{x^9} + 1 \right)
\]

### Step-by-step solution:

1. **As \( x \to \infty \), consider the behavior of each term:**
   - \( x^{15} \): As \( x \to \infty \), \( x^{15} \) grows very large.
   - \( \frac{1}{x^9} \): As \( x \to \infty \), \( \frac{1}{x^9} \) tends to 0 because the denominator grows much faster than the numerator.
   - \( 1 \): This term remains constant.

2. **Dominant term**: 
   As \( x \to \infty \), \( x^{15} \) dominates the expression because it grows much faster than the other terms, and the term \( \frac{1}{x^9} \) approaches 0.

3. **Conclusion**:
   Therefore, the limit is dominated by the \( x^{15} \) term, and we get:

   \[
   \lim_{x \to \infty} \left( x^{15} + \frac{1}{x^9} + 1 \right) = \infty
   \]

### Final Answer:
\[
\lim_{x \to \infty} \left( x^{15} + \frac{1}{x^9} + 1 \right) = \infty
\]

Would you like more details on this? Here are 5 related questions:

1. What happens if the limit is evaluated as \( x \to 0 \) instead of infinity?
2. How does the dominant term affect limits in general?
3. What is the limit behavior of \( \frac{1}{x^n} \) as \( x \to \infty \)?
4. How can L'Hopital's rule apply to limits involving infinity?
5. How do powers of \( x \) affect the growth rates of functions?

**Tip**: Always analyze each term separately when taking limits, focusing on which terms grow faster or slower.

Explore the limit of the expression x^15 + 1/x^9 + 1 as x approaches infinity. This problem highlights key concepts in limit calculation, including dominance of terms and behavior of functions at infinity. Suitable for students in Grades 10-12.

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