18. determine each of the following limit statements for the graph of f(x):\n\n$f(x)=-x^{7}+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\rightarrow-\infty}f(x))

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

Step3: Find (\lim_{x\rightarrow\infty}f(x))

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

Answer:

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