With weird limits like this, a good way to handle them is through series expansion.
For small #absx# we have
#sin x = x -x^3/(3!)+O(x^5)# then
#sinx/x = (x -x^3/(3!)+O(x^5))/x = 1-x^2/(3!) + O(x^4) #
and
#sinx/(x-sinx) = (x - x^3/(3!)+O(x^5))/(x-x+x^3/(3!)-O(x^5))=(1-x^2/(3!) + O(x^4))/(x^2/(3!)-O(x^4))#
and then for small #abs x#
#(sinx/x)^(sinx/(x-sinx)) approx (1-x^2/(3!))^((1-x^2/(3!) +)/(x^2/(3!)))#
now calling #x^2/(3!) = 1/y#
#(sinx/x)^(sinx/(x-sinx)) approx (1-1/y)^((1-1/y)/(1/y)) = (1-1/y)^(y-1)#
now #lim_(x->0)(sinx/x)^(sinx/(x-sinx)) equiv lim_(y->oo)(1-1/y)^(y-1)# and
# lim_(y->oo)(1-1/y)^(y-1) = (lim_(y->oo)(1-1/y)^y)/(lim_(y->oo)(1+1/y)) = e^-1/1 = e^-1#
