y[x_]:=(2x+2)/(3x-4)
Plot[{y[x],2/3},{x,-5,5},PlotRange->{-6,6},PlotPoints->1000]
{Limit[y[x],x->4/3,Direction->1],Limit[y[x],x->4/3,Direction->-1]}
{Limit[y[x],x->-Infinity],Limit[y[x],x->Infinity]}
Vertical asymptote: x = 4/3
Horizontal asymptotes: y = 2/3
Slanted asymptotes: None