Question

describe the end - behavior of the polynomial function using $lim_{x
ightarrowinfty}f(x)$ and $lim_{x
ightarrow-infty}f(x)$. $f(x)=5x^{2}+x^{3}+3x - 2$. describe the end - behavior using limits. select the correct choice below and fill in the answer boxes to complete your choice. a. $lim_{x
ightarrowinfty}f(x)=square$ and $lim_{x
ightarrow-infty}f(x)=square$ because the order of the polynomial, $n = square$, is even and the leading coefficient, $square$, is greater than 0. b. $lim_{x
ightarrowinfty}f(x)=square$ and $lim_{x
ightarrow-infty}f(x)=square$ because the order of the polynomial, $n=square$, is odd and the leading coefficient, $square$, is greater than 0. c. $lim_{x
ightarrowinfty}f(x)=square$ and $lim_{x
ightarrow-infty}f(x)=square$ because the order of the polynomial, $n=square$, is odd and the leading coefficient, $square$, is less than 0. d. $lim_{x
ightarrowinfty}f(x)=square$ and $lim_{x
ightarrow-infty}f(x)=square$ because the order of the polynomial, $n=square$, is even and the

Explanation:

Step1: Identify the leading - term

The polynomial is \(f(x)=5x^{2}+x^{3}+3x - 2\). The leading - term is the term with the highest power of \(x\), which is \(x^{3}\). The degree \(n\) of the polynomial is 3 (since the highest power of \(x\) is 3), and the leading coefficient is 1.

Step2: Recall end - behavior rules for polynomials

For a polynomial \(y = a_{n}x^{n}+a_{n - 1}x^{n - 1}+\cdots+a_{1}x + a_{0}\) with leading term \(a_{n}x^{n}\), if \(n\) is odd and \(a_{n}>0\), then \(\lim_{x
ightarrow\infty}f(x)=\infty\) and \(\lim_{x
ightarrow-\infty}f(x)=-\infty\).

Step3: Calculate the limits

Since the degree \(n = 3\) (odd) and the leading coefficient \(a_{n}=1>0\), we have \(\lim_{x
ightarrow\infty}(5x^{2}+x^{3}+3x - 2)=\infty\) and \(\lim_{x
ightarrow-\infty}(5x^{2}+x^{3}+3x - 2)=-\infty\).

Answer:

B. \(\lim_{x
ightarrow\infty}f(x)=\infty\) and \(\lim_{x
ightarrow-\infty}f(x)=-\infty\) because the order of the polynomial, \(n = 3\), is odd and the leading coefficient, \(1\), is greater than 0.