https://file.aiquickdraw.com/mathbot/file/1726758429969k2fvylck.jpg

The limit shown in the image is:

\[
\lim_{x \to \infty} \frac{9x^3 + 1}{x^2 - 2x + 2}
\]

### Solution:

When evaluating limits as \( x \to \infty \), we typically divide the numerator and the denominator by the highest power of \( x \) found in the denominator to simplify.

In this case, the highest power of \( x \) in the denominator is \( x^2 \). Let's divide both the numerator and the denominator by \( x^2 \):

\[
\lim_{x \to \infty} \frac{\frac{9x^3 + 1}{x^2}}{\frac{x^2 - 2x + 2}{x^2}} = \lim_{x \to \infty} \frac{9x + \frac{1}{x^2}}{1 - \frac{2}{x} + \frac{2}{x^2}}
\]

As \( x \to \infty \):
- \( \frac{1}{x^2} \to 0 \)
- \( \frac{2}{x} \to 0 \)
- \( \frac{2}{x^2} \to 0 \)

So, the expression simplifies to:

\[
\lim_{x \to \infty} \frac{9x + 0}{1 - 0 + 0} = \lim_{x \to \infty} 9x
\]

Since \( 9x \to \infty \), the limit does not converge to a finite value; instead, it **diverges to infinity**.

### Final Answer:

\[
\lim_{x \to \infty} \frac{9x^3 + 1}{x^2 - 2x + 2} = \infty
\]

Would you like more details on any part of this explanation? 

Here are some related questions that might deepen your understanding:
1. How would this limit change if the highest degree term in the denominator was \(x^3\) instead of \(x^2\)?
2. What happens if we were finding the limit as \(x \to -\infty\)?
3. Can you explain why dividing by the highest power of \(x\) helps simplify limits at infinity?
4. What are the different cases when the degree of the numerator is less than, equal to, or greater than the degree of the denominator for limits at infinity?
5. How would this change if the numerator was \(9x^2 + 1\) instead of \(9x^3 + 1\)?

**Tip:** When dealing with limits at infinity, always compare the degrees of the numerator and denominator to predict the behavior as \( x \to \infty \).

Evaluate the limit of (9x^3 + 1) / (x^2 - 2x + 2) as x approaches infinity.

Learn how to find the limit of the rational function (9x^3 + 1) / (x^2 - 2x + 2) as x approaches infinity. This step-by-step solution covers limit theorems and polynomial growth behavior for students in Grades 11-12 or undergraduate levels.

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