We have indeterminateness of type
0/0,
i.e. limit for the numerator is
x→0+limtan(9x)=0and limit for the denominator is
x→0+lim(8x)=0Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
x→0+lim(8xtan(9x))=
Let's transform the function under the limit a few
x→0+lim(8xtan(9x))=
x→0+lim(dxd8xdxdtan(9x))=
x→0+lim(89tan2(9x)+89)=
x→0+lim(89tan2(9x)+89)=
89It can be seen that we have applied Lopital's rule (we have taken derivatives with respect to the numerator and denominator) 1 time(s)