Question

How do you find the slant asymptote of `y=(x^3)/((x^2)-3)`?

Answer 1

The slant asymptote is: `y=x`

Explanation:

Given:

`y = x^3/(x^2-3)`

Note that:

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

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

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

and:

`lim_(x->+-oo) (3x)/(x^2-3) = lim_(x->+-oo) 3/(x-3/x) = lim_(x->+-oo)3/x = 0`

So:

`y = x^3/(x^2-3)`

is asymptotic to `y=x`