Question

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

Explanation:

Step1: Recall end - behavior rules of polynomials

The end - behavior of a polynomial \(f(x)=a_nx^n + a_{n - 1}x^{n-1}+\cdots+a_0\) is determined by the degree \(n\) and the leading coefficient \(a_n\).

Step2: Analyze cases for odd - degree polynomials

If \(n\) is odd:

• When \(a_n>0\), \(\lim_{x

ightarrow-\infty}f(x)=-\infty\) and \(\lim_{x
ightarrow+\infty}f(x)=+\infty\).

• When \(a_n < 0\), \(\lim_{x

ightarrow-\infty}f(x)=+\infty\) and \(\lim_{x
ightarrow+\infty}f(x)=-\infty\).

Step3: Analyze cases for even - degree polynomials

If \(n\) is even:

• When \(a_n>0\), \(\lim_{x

ightarrow-\infty}f(x)=+\infty\) and \(\lim_{x
ightarrow+\infty}f(x)=+\infty\).

• When \(a_n < 0\), \(\lim_{x

ightarrow-\infty}f(x)=-\infty\) and \(\lim_{x
ightarrow+\infty}f(x)=-\infty\).

Answer:

The end - behavior of a polynomial \(y = f(x)\) is determined as follows:
If the degree \(n\) of the polynomial is odd and the leading coefficient is positive, \(\lim_{x
ightarrow-\infty}f(x)=-\infty\) and \(\lim_{x
ightarrow+\infty}f(x)=+\infty\).
If the degree \(n\) of the polynomial is odd and the leading coefficient is negative, \(\lim_{x
ightarrow-\infty}f(x)=+\infty\) and \(\lim_{x
ightarrow+\infty}f(x)=-\infty\).
If the degree \(n\) of the polynomial is even and the leading coefficient is positive, \(\lim_{x
ightarrow-\infty}f(x)=+\infty\) and \(\lim_{x
ightarrow+\infty}f(x)=+\infty\).
If the degree \(n\) of the polynomial is even and the leading coefficient is negative, \(\lim_{x
ightarrow-\infty}f(x)=-\infty\) and \(\lim_{x
ightarrow+\infty}f(x)=-\infty\). Without specific options in a proper multiple - choice format, we can't select a single lettered option, but these are the general rules for end - behavior based on degree and leading coefficient.