https://file.aiquickdraw.com/mathbot/file/17273538820548gf51fuz.jpg

The expression given in the image is:

\[
\lim_{n \to \infty} \frac{n^2 + 4}{9 - 3\left(\frac{n^2 + 1}{n^3 + 5n}\right) + 2}
\]

### Step-by-Step Solution:

#### Step 1: Simplify the denominator
We begin by simplifying the expression in the denominator. Notice the complex term involving \(n^2 + 1\) and \(n^3 + 5n\).

\[
\frac{n^2 + 1}{n^3 + 5n} = \frac{n^2(1 + \frac{1}{n^2})}{n^3(1 + \frac{5}{n^2})}
\]

As \(n \to \infty\), both \(\frac{1}{n^2} \to 0\) and \(\frac{5}{n^2} \to 0\), so the term simplifies to:

\[
\frac{n^2 + 1}{n^3 + 5n} \sim \frac{n^2}{n^3} = \frac{1}{n}
\]

Thus, the expression in the denominator becomes approximately:

\[
9 - 3 \cdot \frac{1}{n} + 2 \quad \text{as} \quad n \to \infty
\]

Simplifying:

\[
9 - 3\left(\frac{1}{n}\right) + 2 = 11 - \frac{3}{n}
\]

#### Step 2: Take the limit as \(n \to \infty\)
As \(n \to \infty\), the term \(\frac{3}{n} \to 0\). So the denominator simplifies to:

\[
11
\]

#### Step 3: Analyze the numerator
The numerator is \(n^2 + 4\). As \(n \to \infty\), this term behaves like \(n^2\).

#### Step 4: Combine the results
The entire expression for the limit becomes:

\[
\lim_{n \to \infty} \frac{n^2 + 4}{11}
\]

As \(n \to \infty\), this simplifies to:

\[
\lim_{n \to \infty} \frac{n^2}{11} = \infty
\]

Thus, the limit is:

\[
\boxed{\infty}
\]

Would you like further details or have any specific questions about this solution?

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Here are some related questions to explore:

1. What happens if the degree of the numerator and denominator are the same?
2. How does the behavior of rational functions change as \(n \to \infty\)?
3. How would the limit change if the term in the numerator were \(n\) instead of \(n^2\)?
4. Can L'Hopital's Rule be applied to this limit? Why or why not?
5. What if the highest-degree term in the denominator had been \(n^4\) instead of \(n^3\)?

**Tip:** When working with limits at infinity, comparing the degrees of terms in the numerator and denominator can help you quickly identify the limit behavior.

lim (n → ∞) of (n^2 + 4) / (9 - 3((n^2 + 1) / (n^3 + 5n)) + 2)

This problem involves finding the limit of a rational function as n approaches infinity. Key steps include simplifying the rational terms and analyzing the behavior of the numerator and denominator at infinity. Suitable for college-level calculus students.

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