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 even and the leading coefficient, $square$, is less 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 odd and the

Explanation:

Step1: Identify the degree and leading coefficient

The polynomial $f(x)=5x^{2}+x^{3}+3x - 2$ can be written in standard form as $f(x)=x^{3}+5x^{2}+3x - 2$. The degree $n = 3$ (the highest - power of $x$) and the leading coefficient is $1$ (the coefficient of $x^{3}$). Since $n = 3$ (odd) and the leading coefficient $a = 1>0$.

Step2: Determine the limit as $x\to\infty$

For a polynomial $y = a_nx^n+\cdots+a_0$ with $n$ odd and $a_n>0$, as $x\to\infty$, we have $\lim_{x\to\infty}(x^{3}+5x^{2}+3x - 2)=\infty$. Because as $x$ gets larger and larger, the term $x^{3}$ dominates the other terms, and since the coefficient of $x^{3}$ is positive, the function value increases without bound.

Step3: Determine the limit as $x\to-\infty$

For a polynomial $y = a_nx^n+\cdots+a_0$ with $n$ odd and $a_n>0$, as $x\to-\infty$, we have $\lim_{x\to-\infty}(x^{3}+5x^{2}+3x - 2)=-\infty$. Because when $x$ is a large negative number, $x^{3}$ is negative (since $n = 3$ is odd), and the function value decreases without bound.

Answer:

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