Find all vertical and horizontal asymptotes of the graph of the given function. $$f ( x ) = \frac { x ^ { 2 } - 3 x } { x ^ { 2 } + 5 x + 6 }$$
A) Vertical asymptotes: x = -3, x = -2; horizontal asymptote: y = 1
B) Vertical asymptotes: x = -3, x = -2; horizontal asymptote: y = 3
C) Vertical asymptotes: x = -3, x = -2, x = 3; horizontal asymptote: y = 0
D) Vertical asymptotes: x = -3, x = -2, x = 3; horizontal asymptote: y = 1

Question

Quizplus · Accepted Answer

Vertical asymptotes occur where the denominator is zero but the numerator is not, so we factor the denominator to find $x^2 + 5x + 6 = (x + 3)(x + 2)$, giving vertical asymptotes at $x = -3$ and $x = -2$. The horizontal asymptote is found by comparing the degrees of the polynomials in the numerator and denominator. Since they are the same (degree 2), the horizontal asymptote is the ratio of the leading coefficients, which is $1/1 = 1$.

Find All Vertical and Horizontal Asymptotes of the Graph of the Given