Find a formula for a function f that satisfies the following conditions.

Question

lim x→±∞ f(x) = 0,   lim x→0 f(x) = −∞,  f(3) = 0, lim x→6⁻ f(x) = ∞,   lim x→6⁺ f(x) = −∞

F(x)=___________

Answer 1

We will assume that this is a polynomial fraction.

The horizontal asymptote is 0 tells us that the degree of the numerator is less than the degree of the denominaor.

lim x -> 0 = -∞ tells us that x^n is in the denominator, where n is even.

f(3) = 0 tells us that (x-3) is a factor in the denominator.

lim x -> 6^- = ∞ and

lim x -> 6⁺ = -∞ tells us that (x-6)^n is in the denominator, n odd

The simplest polynomial meeting these conditions, as it should be negative for x > 6, positive for x < 6 and near,negative on both sides of 0 (meaning a single power of x-3 works), and have numerator degree less than denominator degree (we have 1 and 3 here) is

(3-x)/((x-6)x²) meets all of these conditions - note that I reversed x-3.