https://file.aiquickdraw.com/mathbot/file/1726841599425rkagas0z.jpg

The function given in the image is:

\[
f(n) = 2(n - 3)^2(n + 4)^3(n + 2)^2
\]

We are tasked with analyzing the "long-run behavior" of this function, which means we need to describe the behavior of \( f(n) \) as \( n \) approaches both positive infinity (\( n \to \infty \)) and negative infinity (\( n \to -\infty \)).

### 1. Behavior as \( n \to \infty \):

When \( n \to \infty \), the terms inside the parentheses grow large in a positive direction:
- \( (n - 3)^2 \) behaves like \( n^2 \).
- \( (n + 4)^3 \) behaves like \( n^3 \).
- \( (n + 2)^2 \) behaves like \( n^2 \).

Thus, for large \( n \), \( f(n) \) approximates:

\[
f(n) \approx 2 \cdot n^2 \cdot n^3 \cdot n^2 = 2n^7
\]

Since \( 2n^7 \) dominates for large positive \( n \), the function grows without bound and tends toward positive infinity:

\[
\lim_{n \to \infty} f(n) = \infty
\]

### 2. Behavior as \( n \to -\infty \):

Now consider the behavior as \( n \to -\infty \):
- \( (n - 3)^2 \) behaves like \( n^2 \), but since it's squared, it's always positive.
- \( (n + 4)^3 \) behaves like \( (-n)^3 \), which is negative because of the odd power.
- \( (n + 2)^2 \) behaves like \( n^2 \), and is always positive.

For large negative \( n \), the function behaves like:

\[
f(n) \approx 2 \cdot n^2 \cdot (-n^3) \cdot n^2 = 2(-n^7) = -2n^7
\]

Since \( -2n^7 \) dominates for large negative \( n \), the function becomes increasingly negative and tends toward negative infinity:

\[
\lim_{n \to -\infty} f(n) = -\infty
\]

### Summary:
- As \( n \to \infty \), \( f(n) \to \infty \).
- As \( n \to -\infty \), \( f(n) \to -\infty \).

Would you like more details or further clarifications?

Here are 5 related questions you can explore:
1. How do you find the leading term for polynomials like this one?
2. What happens to the function's behavior near zero or other critical points?
3. How does the degree of the polynomial affect its end behavior?
4. Can you determine the roots or zeros of this function?
5. What is the significance of squaring and cubing terms in the long-run behavior?

**Tip**: When analyzing end behavior, always focus on the highest powers of \( n \), as they dominate the function for large values of \( n \).

Describe the long-run behavior of f(n) = 2(n - 3)^2(n + 4)^3(n + 2)^2 as n → -∞ and n → ∞.

This detailed solution explains how to determine the long-run behavior of f(n) = 2(n - 3)^2(n + 4)^3(n + 2)^2 as n approaches both infinity and negative infinity. Learn how to analyze polynomial end behavior using leading terms and limits.

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