Question

How do you find vertical, horizontal and oblique asymptotes for `G(x)=(3*x^4 +4)/(x^3+3*x)`?

Answer 1

Vertical asymptote at `x_v = 0`
Slant asymptote `y = 3x`

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 =0` and `n = m+1` with `n = 4` and `m = 3`

Slant asymptotes are obtained considering

`(p_n(x))/(p_{n-1}(x)) approx y = a x+x`

for large values of `abs(x)`.
In the present case we have

`(p_n(x))/(p_{n-1}(x)) = (3x^4+4)/(x^3+3x)`

then

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

equating coefficients

`{ (4 - e=0), (-3 b - d=0),( -3 a - c=0),( -b=0),(3 - a=0) :}`

solving for `a,b,c,d,e` we have `{a =3, b = 0, c = -9, d = 0,e=4}`
substituting in `y = a x + b`

`y = 3x`