Lim(x->1) [(x^(1/2)-1)/(x^(1/3)-1)]=
=Lim(x->1) {(x^(1/2)-1)*(x^(1/2)+1)*(x^(2/3)+x^(1/3)+1)}/{(x^(1/3)-1)*(x^(2/3)+x^(1/3)+1)*(x^(1/2)+1)}=