Question

1. determine each of the following limit statements for the graph of f(x):

$f(x)=-x^{7}+x + 1$

$lim_{x
ightarrow-infty}f(x)=$

$lim_{x
ightarrowinfty}f(x)=$

Explanation:

Step1: Analyze the leading - term

The leading - term of the polynomial function \(f(x)=-x^{7}+x + 1\) is \(-x^{7}\). The degree of the polynomial is \(n = 7\) (an odd number) and the leading - coefficient \(a=-1\) (negative).

Step2: Find \(\lim_{x

ightarrow-\infty}f(x)\)
As \(x
ightarrow-\infty\), for the leading - term \(-x^{7}\), when \(x
ightarrow-\infty\), let \(y=-x\), then \(y
ightarrow\infty\) and \(-x^{7}=y^{7}\). So \(\lim_{x
ightarrow-\infty}-x^{7}=\infty\). Since the leading - term dominates as \(x
ightarrow\pm\infty\), \(\lim_{x
ightarrow-\infty}f(x)=\lim_{x
ightarrow-\infty}(-x^{7}+x + 1)=\infty\).

Step3: Find \(\lim_{x

ightarrow\infty}f(x)\)
As \(x
ightarrow\infty\), the leading - term \(-x^{7}\) goes to \(-\infty\) because the leading coefficient is negative and the degree is odd. So \(\lim_{x
ightarrow\infty}f(x)=\lim_{x
ightarrow\infty}(-x^{7}+x + 1)=-\infty\).

Answer:

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