Formula for a rational function

Question

Give a formula for a rational function f(x) that satisfies:

• f(2) is undefined

• limx→(2) f(x) = 0

• lim x→(−3+) f(x) = ∞

• lim x→(−3−) f(x) = −∞

Your formula can have numerator and denominator factored (you don’t have to ”multiply them out.”) Show that your formula satisfies all the conditions.

Answer 1

1. Because f(2) is undefined but f(x) has a limit as x -->2, f(x) has a hole at x = 2. Therefore, both the numerator and denominator have the factor (x - 2)
2. Because the limit of f(x) as x --> 2 = 0, there is an additional factor of (x - 2) in the numerator.
3. You determine the limit as x approaches the hole after cancelling the factors that cause the hole.
4. After these factors are cancelled, there must still be a factor in the numerator that creates this limit when x = 2, i.e. an additional (x - 2) must remain in the numerator after cancelling the factors for the hole.
5. Because f(x) goes to +- infinity on either side of x = -3, there must be a factor of (x + 3) in the denominator to create this vertical asymptote.
6. So far we f(x) = a (x - 2)² / (( x + 3)(x - 2))
7. By putting in a test value of x = -4 and x = -2, you can find that "a" must be negative so that the function goes to negative infinity as x -->-3^- and goes to positive infinity as x --> -3⁺

Final answer: f(x) = -(x - 2)² / (( x + 3)(x - 2))