Which of the following functions has a graph with exactly two vertical asymptotes and one horizontal asymptote

Question

a. f(x) = (x + 1)/(2x + 4)

b. g(x) = 2x/(x2 - 4)

c. h(x) = (x2 + 4x + 15)/(x + 2)

d. F(x) = (x3 - x - 1)/[(x - 1)(x + 2)]

e. G(x) = (x3 )/(x - 4)

f. none of these

Answer 1

A function with a vertical asymptote approaches infinity as x approaches some point which generally arises when there is division by 0 at the point x approaches. So a function with 2 vertical asymptotes will have division by 0 at 2 points so we look for a function which has a quadratic term with 2 real roots in the denominator.

(b) and (d) have quadratics in their denominators with 2 distinct real roots so they each have 2 vertical asymptotes. However, (d) does not have a horizontal asymptote since

lim F(x) = ±∞
x->±∞

(b) has 1 horizontal asymptote since

lim g(x) = 0
x->±∞

So (b) has 2 vertical asymptotes at x=±2 and a horizontal asymptote at x=0

Answer 2

b) x=2 and x=-2 are the vertical aysimptotes and y=0 the horizontal one