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.