>>861 x→+0 として

sin(x) = x - (1/6)x^3 + O(x^5)
x*log(sin(x)/x) = x*log(1 - (1/6)x^2 + O(x^4)) = -(x^3/6) + O(x^5)
(sin(x)/x)^x = 1 - (1/6)x^3 + O(x^5)
sin(x)^x = x^x(1 - (1/6)x^3 + O(x^5))

x^x も展開するなら
x^x = e^(x*log(x)) = 1 + x*log(x) + (1/2)(x*log(x))^2 + (1/6)(x*log(x))^3 + O((x*log(x))^4)

∴ sin(x)^x
= 1 + x*log(x) + (1/2)(x*log(x))^2 + (1/6)(x*log(x))^3 - (1/6)x^3 + O((x*log(x))^4)