Question

How do you find vertical, horizontal and oblique asymptotes for `f(x)= (6x^4-x^3+6x+5)/(x^3+1)`?

Answer 1

Vertical asypmtote: `x=-1`
Oblique asymptote: `y=6x-1`

Explanation:

The vertical asymptote occurs at the point where the denominator is equal to zero, and in this case, `x=-1` satisfies `x^3+1=0`. Since the degree of the numerator is larger than the degree of the denominator, there is an oblique asymptote instead of a horizontal asymptote. Using polynomial long division, we find that `f(x)=6x-1+6/(x^3+1)`, so the oblique asyptote is `y=6x-1`.