Question

How do you find all the asymptotes for function `f(x) = (x+4)/(3x^2+5x-2)`?

Answer 1

The vertical asymptotes are `x=1/3` and `x=-2`
The vertical asymptote is `y=0`
There are no slant asymptotes

Explanation:

Let's factorise the denominator `3x^2+5x-2 =(3x-1)(x+2)`

As we cannot divide by `0`, the vertical asymptotes are `x=1/3` and `x=-2`

The degree of the numerator is `<` the degree of the denominator, so we don't have a slant asymptote.

`lim_(x->-oo)f(x)=lim_(x->-oo)x/(3x^2)=lim_(x->-oo)1/(3x)=0^(-)`

`lim_(x->+oo)f(x)=lim_(x->+oo)x/(3x^2)=lim_(x->+oo)1/(3x)=0^(+)`

So `y=0` is a horizontal asymptote
graph{(x+4)/(3x^2+5x-2) [-11.05, 6.73, -6.2, 2.69]}