lim(1+3x)^(1/x-2)当x趋于0时原式极限是多少?

(x->0)lim(1+3x)^(1/x-2)
=(x->0)lim(1+3x)^(1/x)×(x->0)lim(1+3x)^(-2)
=(x->0)lim(1+3x)^(1/x)×1
=(x->0)lim(1+3x)^(1/x)
=[(x->0)lim(1+3x)^(1/3x)]³
=e³
这里用到了
x^(a+b)=x^a×x^b
x^ab=(x^a)^b
(x->0)lim(1+x)^(1/x)=e