Question

How do you find vertical, horizontal and oblique asymptotes for `f(x) =( x^3 - 4x^2 + 5x + 5)/(x - 1)`?

Answer 1

`f(x)` has one vertical asymptote `x=1` and no other linear asymptotes.

Explanation:

`f(x) = (x^3-4x^2+5x+5)/(x-1)`

`color(white)(f(x)) = (x^3-x^2-3x^2+3x+2x-2+7)/(x-1)`

`color(white)(f(x)) = x^2-3x+2 + 7/(x-1)`

This function has a vertical asymptote `x=1`.

As `x->+-oo` we have `7/(x-1)->0`, so `f(x)` is asymptotic to `x^2-3x+2`, which is not a line.

So there are no other linear asymptotes.

graph{(x^3-4x^2+5x+5)/(x-1) [-10, 10, -200, 200]}