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:
We are asked to evaluate the behavior of as and as .
1. As :
- For large values of , the highest degree term dominates the behavior of the polynomial. In this case, the highest degree term is .
- As , grows rapidly, and thus the term will dominate over the others, including the constant term .
- Therefore, as , .
2. As :
- Again, the highest degree term dominates the behavior. But because the exponent is odd, will be negative for negative values of .
- As , because will approach negative infinity.
- Therefore, as , .
Final Answers:
- As , .
- As , .
Would you like further details or explanations? Here are five related questions to expand your understanding:
- What is the general approach to analyzing limits of polynomial functions at infinity?
- How do different powers of affect the growth rate of a polynomial?
- What is the difference in behavior between even and odd degree terms in polynomials?
- How do we determine the dominant term in a polynomial for large values of ?
- How would the behavior of change if the degree of the polynomial were even?
Tip: When analyzing polynomials for large , 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
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