Question

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

Answer 1

Vertical asymptotes at `x_v = {2,-2}`
Slant asymptote `y = x`

Explanation:

In a polynomial fraction `f(x) = (p_n(x))/(p_m(x))` we have:

`1)` vertical asymptotes for `x_v` such that `p_m(x_v)=0`
`2)` horizontal asymptotes when `n le m`
`3)` slant asymptotes when `n = m + 1`
In the present case we have `x_v = {-2, 2}` and `n = m+1` with `n = 3` and `m = 2`

Slant asymptotes are obtained considering `(p_n(x))/(p_{n-1}(x)) approx y = a x+b` for large values of `abs(x)`

In the present case we have

`(p_n(x))/(p_{n-1}(x)) = x^3/(x^2-4)`
`p_n(x)=p_{n-1}(x)(a x+b)+r_{n-2}(x)`
`r_{n-2}(x)=c x + d`
`x^3 = (x^2-4)(a x + b) + c x + d`

equating coefficients

`{ (4 b - d=0), (4 a - c=0), (-b=0), (1 - a=0) :}`

solving for `a,b,c,d` we have `{a = 1, b = 0, c = 4, d = 0}`
substituting in `y = a x + b`

`y = x`