We have indeterminateness of type
oo/oo,
i.e. limit for the numerator is
n→∞lim(n+1)=∞and limit for the denominator is
n→∞lim(n+2)=∞Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
n→∞lim(n+2n+1)=
n→∞lim(dnd(n+2)dnd(n+1))=
n→∞lim1=
n→∞lim1=
1It can be seen that we have applied Lopital's rule (we have taken derivatives with respect to the numerator and denominator) 1 time(s)