A limit which I don't understand?

Question

lim x--> 0 (tanx/x)^(1/x^2)
Transforming it into lim x-->0 e^(1/x^2)*ln (tanx/x) only gives an impossible ln 0/0, in this case I can't really do much.
Well,I'll still try to solve this ,any help from you guys is much appreciated.

Answer 1

Lim (tanx/x)^(1/x^2) = lim [1+tanx/x-1]^(1/x^2=
lim[(1+tanx/x-1)^(1/(tanx/x-1)
raised to (tanx/x-1)*1/x^2)
The expression in[ ]^(1/(tanx/x-1) has limit e
so we need to calculate
lim (tanx-x)/x^3.
This can be done using L´Hôpital Mc Laurins
Bi the first =lm (1/cos^2 -1)/3x^2 = lim 1-cos^2x)/3x^2 as cosx =>1
=2cosx*sinx/6x and as we shuold know thar six/x=>1 the limit is 1/3
so the searched limit is e^1/3