The given function is:

$f(n) = 3n^7 + 2n^4 - 2n^2 - 3$

We are asked to evaluate the behavior of $f(n)$ as $n \to \infty$ and as $n \to -\infty$.

1. As $n \to \infty$:

• For large values of $n$, the highest degree term dominates the behavior of the polynomial. In this case, the highest degree term is $3n^7$.
• As $n \to \infty$, $n^7$ grows rapidly, and thus the term $3n^7$ will dominate over the others, including the constant term $-3$.
• Therefore, as $n \to \infty$, $f(n) \to \infty$.

2. As $n \to -\infty$:

• Again, the highest degree term $3n^7$ dominates the behavior. But because the exponent $7$ is odd, $n^7$ will be negative for negative values of $n$.
• As $n \to -\infty$, $3n^7 \to -\infty$ because $n^7$ will approach negative infinity.
• Therefore, as $n \to -\infty$, $f(n) \to -\infty$.

Final Answers:

• As $n \to \infty$, $f(n) \to \infty$.
• As $n \to -\infty$, $f(n) \to -\infty$.

Would you like further details or explanations? Here are five related questions to expand your understanding:

1. What is the general approach to analyzing limits of polynomial functions at infinity?
2. How do different powers of $n$ affect the growth rate of a polynomial?
3. What is the difference in behavior between even and odd degree terms in polynomials?
4. How do we determine the dominant term in a polynomial for large values of $n$?
5. How would the behavior of $f(n)$ change if the degree of the polynomial were even?

Tip: When analyzing polynomials for large $n$, always focus on the term with the highest degree, as it will dominate the behavior of the function.