Math  /  Calculus

QuestionDescribe the long run behavior of f(n)=3n84n3+n2+5f(n)=3 n^{8}-4 n^{3}+n^{2}+5 As n,f(n)?n \rightarrow-\infty, f(n) \rightarrow ? \vee As n,f(n)?n \rightarrow \infty, f(n) \rightarrow ? \vee

Studdy Solution

STEP 1

1. The function f(n)=3n84n3+n2+5f(n) = 3n^8 - 4n^3 + n^2 + 5 is a polynomial function.
2. The behavior of polynomial functions as nn \rightarrow \infty or nn \rightarrow -\infty is primarily determined by the highest degree term.
3. For large values of n|n|, the highest degree term 3n83n^8 will dominate the behavior of the function.

STEP 2

1. Identify the highest degree term in the polynomial.
2. Analyze the behavior of the highest degree term as nn \rightarrow \infty.
3. Analyze the behavior of the highest degree term as nn \rightarrow -\infty.
4. Conclude the long-run behavior of f(n)f(n) based on the behavior of the highest degree term.

STEP 3

Identify the highest degree term in the polynomial f(n)=3n84n3+n2+5f(n) = 3n^8 - 4n^3 + n^2 + 5.
The highest degree term is 3n83n^8.

STEP 4

Analyze the behavior of the highest degree term as nn \rightarrow \infty.
As nn \rightarrow \infty, 3n83n^8 \rightarrow \infty because the coefficient 33 is positive and the exponent 88 is even.

STEP 5

Analyze the behavior of the highest degree term as nn \rightarrow -\infty.
As nn \rightarrow -\infty, 3n83n^8 \rightarrow \infty because the exponent 88 is even, making n8n^8 positive, and the coefficient 33 is positive.

STEP 6

Conclude the long-run behavior of f(n)f(n) based on the behavior of the highest degree term.
As nn \rightarrow \infty, f(n)f(n) \rightarrow \infty. As nn \rightarrow -\infty, f(n)f(n) \rightarrow \infty.

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