Question

Does the horizontal asymptote exist or not?

Answer 1

The horizontal asymptote occurs when the graph tries to approach `y = "something"` but never reaches it. The limit as `x` gets large would allow this:

`lim_(x->oo) (x^3 - 2x^2 + 3)/(x-2)`

As `x` gets large, the lower-order terms go away and:

`lim_(x->oo) (x^3 - cancel(2x^2 + 3)^"small")/(x-cancel(2)^"small") = x^3/x = x^2`

So the graph is shaped like `x^2` and has no horizontal asymptote. I superimposed `x^2` on here:

graph{((x^3 - 2x^2 + 3)/(x-2) - y)(x^2 - y) = 0 [-20, 20, -50, 150]}

It does have a vertical asymptote at `x = 2` though, where `y` does not exist.