Question

the function (f) is given by (f(x)=-2x^{7}+5x^{4}+6x^{2}-3). which of the following correctly describes the end - behavior of (f) as the input values increase without bound? a (lim_{x
ightarrow+infty}f(x)=infty) b (lim_{x
ightarrow+infty}f(x)=-infty) c (lim_{x
ightarrow-infty}f(x)=infty) d (lim_{x
ightarrow-infty}f(x)=-infty)

Explanation:

Step1: Identify the leading - term

The function is $f(x)=-2x^{7}+5x^{4}+6x^{2}-3$. The leading - term is $-2x^{7}$ since it has the highest degree ($n = 7$).

Step2: Analyze the end - behavior for $x\to+\infty$

For a polynomial function $y = a_{n}x^{n}+a_{n - 1}x^{n - 1}+\cdots+a_{0}$, when $n$ is odd and $a_{n}<0$, as $x\to+\infty$, $y\to-\infty$. Here, $n = 7$ (odd) and $a_{n}=-2<0$. So, $\lim_{x\to+\infty}f(x)=-\infty$.

Answer:

B. $\lim_{x\to+\infty}f(x)=-\infty$