g(x)=(2x^3+x^2-18x-9)/(x^2+x-6)
A. Find the vertical asymptotes.
b. find any holes.
c. find any oblique or horizontal asymptotes.
d. find if the graph crosses the oblique or horizontal asymptote.
g(x) = (x-3)(x+3)(2x-1)/(x+3)(x-2)
So, while the denominator is zero at x = -3, so is the numerator, so there is a hole there.
The only vertical asymptote is at x=2.
considering only the highest powers, as x gets huge, g(x) approaches 2x^3/x^2 = 2x, which is the slant asymptote.
to cross the oblique asymptote, we need
g(x) = 2x
(2x^3+x^2-18x-9)/(x^2+x-6) = 2x
2x^3+x^2-18x-9 = 2x^3+2x^2-12x
x^2+6x+9=0
(x+3)^2 = 0
x = -3
but that is a hole in the graph, so the curve never intersects the asymptote.