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