Question

高等数学小问题，求解，谢谢，答案有点看不懂

红笔从那一步到下一步怎么来的?

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Answer 1

x->0
(a1)^x = 1+(lna1)x +o(x)
(a2)^x = 1+(lna2)x +o(x)
...
(an)^x = 1+(lnan)x +o(x)
[(a1)^x+(a2)^x+...+(an)^x]/n = 1 +(1/n)[ln(a1.a2...an)]x +o(x)
lim(x->0) { [(a1)^x+(a2)^x+...+(an)^x]/n } ^(1/x)
=lim(x->0) { 1 +(1/n)[ln(a1.a2...an)]x }^(1/x)
= e^{ (1/n)[ln(a1.a2...an)] }
= e^{ ln [(a1.a2...an) ^(1/n)] }
=(a1.a2...an) ^(1/n)
//
consider
lim(x->0) ( 1+ kx)^(1/x) = e^k
k = (1/n)[ln(a1.a2...an)]
=>
=lim(x->0) [1 +(1/n)[ln(a1.a2...an)]x ]^(1/x)
= e^{ (1/n)[ln(a1.a2...an)] }