Question

consider the polynomial function f(n)=−5n^9−n^8−3n^7−4. as n→−∞, f(n)→? as n→∞, f(n)→? question help: video submit question

Explanation:

Step1: Identify the leading - term

The leading - term of the polynomial function $f(n)=-5n^{9}-n^{8}-3n^{7}-4$ is $-5n^{9}$. The degree of the polynomial is $9$ (an odd number) and the leading - coefficient is $-5$ (negative).

Step2: Analyze $n\to-\infty$

When $n\to-\infty$, for the leading - term $y = - 5n^{9}$, since the exponent $9$ is odd, when $n$ is negative, $n^{9}$ is negative. And multiplying by the negative leading - coefficient $-5$, we have $-5n^{9}\to+\infty$ as $n\to-\infty$. As $n\to-\infty$, the leading - term dominates the behavior of the polynomial, so $f(n)\to+\infty$.

Step3: Analyze $n\to\infty$

When $n\to\infty$, for the leading - term $y=-5n^{9}$, since the exponent $9$ is odd, $n^{9}$ is positive. Multiplying by the negative leading - coefficient $-5$, we get $-5n^{9}\to-\infty$ as $n\to\infty$. As $n\to\infty$, the leading - term dominates the behavior of the polynomial, so $f(n)\to-\infty$.

Answer:

As $n\to-\infty$, $f(n)\to+\infty$; As $n\to\infty$, $f(n)\to-\infty$