Question

find a formula for a function f that satisfies the following conditions.
lim(x→±∞) f(x) = 0, lim(x→0) f(x) = -∞, f(2) = 0
lim(x→8⁻) f(x) = ∞, lim(x→8⁺) f(x) = -∞
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 \(\lim_{x
ightarrow0}f(x)=-\infty\), \(\lim_{x
ightarrow8^{-}}f(x)=\infty\), and \(\lim_{x
ightarrow8^{+}}f(x)=-\infty\) 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 \(\lim_{x
ightarrow\pm\infty}f(x)=0\) 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:

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