We need
a^2-b^2=(a+b)(a-b)a2−b2=(a+b)(a−b)
We factorise the denominator
x^2-9=(x+3)(x-3)x2−9=(x+3)(x−3)
y=x/((x+3)(x-3))y=x(x+3)(x−3)
As we cannot divide by 00, x!=3 and x!=3x≠3
The vertical asymptotes are x=-3x=−3 and x=3x=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)xcolor(white)(aaaa)-oocolor(white)(aaaa)-3color(white)(aaaaaaaa)0color(white)(aaaaaaa)+3color(white)(aaaaaaa)+oo
color(white)(aaaa)x+3color(white)(aaaa)-color(white)(aaa)||color(white)(aaaa)+color(white)(aaaa)+color(white)(aaaaa)||color(white)(aaa)+
color(white)(aaaa)xcolor(white)(aaaaaaaa)-color(white)(aaa)||color(white)(aaaa)-color(white)(aaaa)+color(white)(aaaaa)||color(white)(aaa)+
color(white)(aaaa)x-3color(white)(aaaa)-color(white)(aaa)||color(white)(aaaa)-color(white)(aaaa)-color(white)(aaaaa)||color(white)(aaa)+
color(white)(aaaa)ycolor(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]}
