Question

Describe the long-term behavior of f(x)= (3x^3+2x-8)/(x^2+x)

a. horizontal asymptote at y=3

b. Slant asymptote with a slope of 3

c. Slant asymptote at y= 3x

d. none of the above

Answer 1

check the picture below.

so that's the graph of that rational equation.

clearly it doesn't have an asymptote at y = 3, it does have an slant asymptote, notice the line on the right-side is slanted about some asymptotic border, well, if we do long division on the rational, the quotient is the equation of the slant asymptote.

it just so happen the long division gives us a quotient of 3x - 3, with a remainder of 5x - 8.

so it has a slant asymptote, but is at y = 3x - 3.

so I take is none of the above.