y[x_]:=(x-1)/(x^2-9)
Plot[y[x],{x,-5,5},PlotRange->{-5,5},PlotPoints->1000]
{Limit[y[x],x->-3,Direction->1],Limit[y[x],x->-3,Direction->-1]}
{Limit[y[x],x->3,Direction->1],Limit[y[x],x->3,Direction->-1]}
{Limit[y[x],x->-Infinity],Limit[y[x],x->Infinity]}
Vertical asymptotes: x = -3 and x = 3
Horizontal asymptotes: y = 0
Slanted asymptotes: None