高等数学当x趋向于无穷大时,lim[((a1)^(1/x)+(a2)^(1/x)+(a3)^(1/x)+…+(an)^(1/x))/n]^nx

Question

高等数学当x趋向于无穷大时,lim[ ( (a1)^(1/x)+(a2)^(1/x)+(a3)^(1/x)+…+(an)^(1/x) )/n]^nx

Answer 1

令t=1/x,原极限=limx-->0[ ( (a1)^t+(a2)^t+(a3)^t+…+(an)^t)/n]^n/t
=exp{limx-->0 (n/t)ln[ ( (a1)^t+(a2)^t+(a3)^t+…+(an)^t)/n]}
应用诺必达=exp{limx-->0n[((a1)^t ln(a1)+(a2)^t ln(a2)+…+(an)^t ln(an))/((a1)^t+(a2)^t+…+(an)^t)]}
∵limx-->0(ai)^t=1；∴原极限=exp{ln(a1)+ln(a2)+ln(a3)+...ln(an)}
=(a1)(a2)(a3)...(an)
纯手打,