Question

How do you find the vertical, horizontal and slant asymptotes of: `f(x)= ( x^3-2x-3) /( x^2+2)`?

Answer 1

There is only a slant asymptote `y=x`

Explanation:

The domain of the function is `RR` as the denominator is always `>0`
So there is no vertical asymptote

The degree of the numerator is `>` degree of denominator, so we expect a slant asymptote

A long division gives

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

So the slant asymptote is `y=x`
For horizontal asymptotes, we look at the limit of `f(x)` as `x->+-oo`

limit `f(x)` as `x->+oo` is `+oo`

limit `f(x)` as `x->-oo` is `-oo`
So there is no horizontal asymptote

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