Math Problem Statement

Consider the polynomial function f(n) = 3n^7 + 2n^4 - 2n^2 - 3. As n → ∞, f(n) → ?. As n → -∞, f(n) → ?.

Solution

The given function is:

f(n)=3n7+2n42n23f(n) = 3n^7 + 2n^4 - 2n^2 - 3

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

1. As nn \to \infty:

  • For large values of nn, the highest degree term dominates the behavior of the polynomial. In this case, the highest degree term is 3n73n^7.
  • As nn \to \infty, n7n^7 grows rapidly, and thus the term 3n73n^7 will dominate over the others, including the constant term 3-3.
  • Therefore, as nn \to \infty, f(n)f(n) \to \infty.

2. As nn \to -\infty:

  • Again, the highest degree term 3n73n^7 dominates the behavior. But because the exponent 77 is odd, n7n^7 will be negative for negative values of nn.
  • As nn \to -\infty, 3n73n^7 \to -\infty because n7n^7 will approach negative infinity.
  • Therefore, as nn \to -\infty, f(n)f(n) \to -\infty.

Final Answers:

  • As nn \to \infty, f(n)f(n) \to \infty.
  • As nn \to -\infty, f(n)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 nn 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 nn?
  5. How would the behavior of f(n)f(n) change if the degree of the polynomial were even?

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

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Math Problem Analysis

Mathematical Concepts

Polynomials
Limits
Asymptotic Behavior

Formulas

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

Theorems

Dominant term theorem for polynomials

Suitable Grade Level

Grades 11-12