We have indeterminateness of type
oo/oo,
i.e. limit for the numerator is
x→∞lim(log(x)+log(3))=∞and limit for the denominator is
x→∞limx2=∞Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
x→∞lim(x2log(3x))=
x→∞lim(dxdx2dxd(log(x)+log(3)))=
x→∞lim(2x21)=
x→∞lim(2x21)=
0It can be seen that we have applied Lopital's rule (we have taken derivatives with respect to the numerator and denominator) 1 time(s)