①ｌｉｍ　ｘ→０　(ａ＾x-1)/x (ａ＞０）
(i)a=1の時 代入して0
(ii)a≠1の時
\(a^{x}\)-1=tとしてx=log(t+1)/log(a) (a＞0 log(a)≠0)
またx→0の時t→0
(ａ＾x-1)/x = t*log(a)/log(t+1)
= log(a) / (\(\frac{1}{t}\))*log(t+1)
= log(a) / log((t+1)^(\(\frac{1}{t}\)))
\(\vec{lo}\)g(a) / log(e) (t→0)
= log(a)
これは(i)も満たす

②ｌｉｍ　ｘ→０　（２ｓｉｎｘ－ｓｉｎ２ｘ）／ｘ＾３
sin(x)/x→1 (x→0)　・・・(1)
2sin(x)-sin(2x) = 2sin(x)-2sin(x)cos(x)
= 2sin(x)*2*(1-cos(x))/2
= 4sin(x)*(sin(\(\frac{x}{2}\))\()^{2}\)
これを代入して
(2sin(x)-sin(2x))/\(x^{3}\)
=4(sin(x)/x)*(sin(\(\frac{x}{2}\))/(\(\frac{x}{2}\))\()^{2}\)*(\(\frac{1}{4}\))
(1)より
　→1　(x→1)