Question

微积分，这道题怎么做

Answer 1

转换为函数极限，令t=1/n，则
I=lim[t->0]((a^t+b^t+c^t)/3)^(1/t)
lnI
=lim[t->0]ln((a^t+b^t+c^t)/3)/t
=lim[t->0](a^tlna+b^tlnb+c^tlnc)/(a^t+b^t+c^t)（洛必达法则）
=(lna+lnb+lnc)/3
=ln(abc)^(1/3)
所以I=(abc)^(1/3)

Answer 2

I = lim(n->∞) {[ a^(1/n) +b^(1/n) + c^(1/n) ]/3}^n
consider
I = lim(x->∞) {[ a^(1/x) +b^(1/x) + c^(1/x) ]/3}^x
lnL =lim(x->∞) ln{[ a^(1/x) +b^(1/x) + c^(1/x) ]/3} /x (∞/∞)
=lim(x->∞) -3[ lna .a^(1/x)+lnb .b^(1/x)+lnc. c^(1/x) ]
/ {x^2.[ a^(1/x) +b^(1/x) + c^(1/x) ] }
=0
L = e^0 =1