We have indeterminateness of type
oo/oo,
i.e. limit for the numerator is
x→∞limex=∞and limit for the denominator is
x→∞limx2=∞Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
x→∞lim(x2ex)=
Let's transform the function under the limit a few
x→∞lim(x2ex)=
x→∞lim(dxdx2dxdex)=
x→∞lim(2xex)=
x→∞lim(dxdxdxd2ex)=
x→∞lim(2ex)=
x→∞lim(2ex)=
∞It can be seen that we have applied Lopital's rule (we have taken derivatives with respect to the numerator and denominator) 2 time(s)