求极限lim(x->0){3e^[x/(x-1)]-2}^(1/x),

Question

求极限lim(x->0){3e^[x/(x-1)]-2}^(1/x),

Answer 1

令 t= x/(x-1),1/x= 1 - 1/t ,Limit [ x/(x-1),x->0] =0
原式 = Limit [ (3 e^t -2 ) ^ (1-1/t ),t->0]
= Limit [ ( 1 + 3 (e^t -1) ) ^ (1-1/t ),t->0]
e^t -1 t
上式 = Limit [ ( 1 + 3 t ) ^ (1-1/ t ),t->0]
= Limit [ ( 1 + 3 t ) / (1+3t)^(1/ t ),t->0]
= 1/ e^3 = e^(-3)