Question

the function f(x) is defined below. what is the end - behavior of f(x)? f(x)=-1610x + 55x^4+9800 - 51x^3-179x^2 + 3x^5. the degree of the polynomial is, and the leading coefficient is. choose which pair of limits below match the end - behavior: lim_(x→-∞)f(x)=-∞, lim_(x→∞)f(x)=-∞; lim_(x→-∞)f(x)=∞, lim_(x→∞)f(x)=-∞; lim_(x→-∞)f(x)=-∞, lim_(x→∞)f(x)=∞; lim_(x→-∞)f(x)=∞, lim_(x→∞)f(x)=∞

Explanation:

Step1: Identify the degree of the polynomial

The polynomial \(f(x)= - 1610x+55x^{4}+9800 - 51x^{3}-179x^{2}+3x^{5}\) has the highest - power of \(x\) as \(5\). So the degree of the polynomial is \(5\) (an odd number).

Step2: Recall end - behavior rules for polynomials

For a polynomial \(y = a_nx^n+\cdots+a_0\) with leading coefficient \(a_n\) and degree \(n\): If \(n\) is odd and \(a_n>0\), \(\lim_{x
ightarrow\infty}f(x)=\infty\) and \(\lim_{x
ightarrow-\infty}f(x)=-\infty\). If \(n\) is odd and \(a_n < 0\), \(\lim_{x
ightarrow\infty}f(x)=-\infty\) and \(\lim_{x
ightarrow-\infty}f(x)=\infty\). In the polynomial \(f(x)=3x^{5}+55x^{4}-51x^{3}-179x^{2}-1610x + 9800\), the leading coefficient \(a_n = 3>0\) and \(n = 5\) (odd). So \(\lim_{x
ightarrow\infty}f(x)=\infty\) and \(\lim_{x
ightarrow-\infty}f(x)=-\infty\).

Answer:

\(\lim_{x
ightarrow\infty}f(x)=\infty\), \(\lim_{x
ightarrow-\infty}f(x)=-\infty\)