Question

What are the asymptotes of `y=x/(x^2-9)` and how do you graph the function?

Answer 1

The vertical asymptotes are `x=-3` and `x=3`
The horizontal asymptote is `y=0`
No oblique asymptote

Explanation:

We need

`a^2-b^2=(a+b)(a-b)`

We factorise the denominator

`x^2-9=(x+3)(x-3)`

`y=x/((x+3)(x-3))`

As we cannot divide by `0`, x!=3 and `x!=3`

The vertical asymptotes are `x=-3` and `x=3`

There are no oblique asymptotes as the degree of the numerator is `<` than the degree of the denominator

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

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

The horizontal asymptote is `y=0`

We can build a sign chart to have a general view of the graph

`color(white)(aaaa)` `x` `color(white)(aaaa)` `-oo` `color(white)(aaaa)` `-3` `color(white)(aaaaaaaa)` `0` `color(white)(aaaaaaa)` `+3` `color(white)(aaaaaaa)` `+oo`

`color(white)(aaaa)` `x+3` `color(white)(aaaa)` `-` `color(white)(aaa)` `||` `color(white)(aaaa)` `+` `color(white)(aaaa)` `+` `color(white)(aaaaa)` `||` `color(white)(aaa)` `+`

`color(white)(aaaa)` `x` `color(white)(aaaaaaaa)` `-` `color(white)(aaa)` `||` `color(white)(aaaa)` `-` `color(white)(aaaa)` `+` `color(white)(aaaaa)` `||` `color(white)(aaa)` `+`

`color(white)(aaaa)` `x-3` `color(white)(aaaa)` `-` `color(white)(aaa)` `||` `color(white)(aaaa)` `-` `color(white)(aaaa)` `-` `color(white)(aaaaa)` `||` `color(white)(aaa)` `+`

`color(white)(aaaa)` `y` `color(white)(aaaaaaaa)` `-` `color(white)(aaa)` `||` `color(white)(aaaa)` `+` `color(white)(aaaa)` `-` `color(white)(aaaaa)` `||` `color(white)(aaa)` `+`

The intercepts are `(0,0)`

`lim_(x->-3^-)y=-oo`

`lim_(x->-3^+)y=+oo`

`lim_(x->3^-)y=-oo`

`lim_(x->3^+)y=+oo`

Here is the graph

graph{(y-(x)/(x^2-9))(y)(y-1000(x+3))(y-1000(x-3))=0 [-18.05, 18.02, -9.01, 9.03]}