A rational function f(x) contains quadratic functions in both the numerator and denominator. Also the function f(x) has a vertical asymptote at x=5, a single x intercept of (2,0) and f is removeably discontinuous at x=1 because the lim x-> 1 is -1/9.

Question

Find f(0) and lim x-> infinity of F(x)

Answer 1

f(1) = 0/0, so (x-1) must be a factor of both the numerator and denominator.

f(5) = 0, So (x-5) is a factor of the denominator only.

f(2)=0, so (x-2) is a factor of the numerator only.

f(x) = a(x-1)(x-2)/(x-1)(x-5)

for x≠1 f(x) = a(x-2)/(x-5)
as x->1, f(x) -> -1/9, so
a(-1)/(-4) = a(1/4) = -1/9
a = -4/9

f(x) = -4/9 (x^2-3x+2)/(x^2-6x+5)