find a formula for a function f that satisfies the following conditions.\nlim(x→±∞) f(x) = 0, lim(x→0) f(x)…

Explanation:

Step1: Analyze the zero - value condition

Since (f(2)=0), ((x - 2)) is a factor of the numerator of the rational - function form of (f(x)).

Step2: Analyze the vertical asymptote conditions

The limits (\lim_{x\rightarrow0}f(x)=-\infty), (\lim_{x\rightarrow8^{-}}f(x)=\infty), and (\lim_{x\rightarrow8^{+}}f(x)=-\infty) imply that (x = 0) and (x = 8) are vertical asymptotes. So, ((x-0)=x) and ((x - 8)) are factors of the denominator.

Step3: Analyze the horizontal asymptote condition

The limit (\lim_{x\rightarrow\pm\infty}f(x)=0) indicates that the degree of the numerator is less than the degree of the denominator. A simple form that satisfies all these conditions is a rational function. Let (f(x)=\frac{x - 2}{x(x - 8)}).

Answer:

(f(x)=\frac{x - 2}{x(x - 8)})