Question

高数题目求解答？

Answer 1

答案是√30

追答

分离分子分母，化成下面说的形式。

用结论(1+x)的1/x次方等于e

追问

可以具体一点嘛

Answer 2

lim(n->∞) { [ 2^(1/n) + 3^(1/n) +5^(1/n) ]/3 }^n
consider
y->0+
2^y = 1+(ln2)y +o(y)
3^y = 1+(ln3)y +o(y)
5^y = 1+(ln5)y +o(y)
(2^y + 3^y +5^y) /3 = 1+ (1/3)ln(30)y + o(y)
lim(x->∞) { [ 2^(1/x) + 3^(1/x) +5^(1/x) ]/3 }^x
y=1/x
lim(y->0+) [ (2^y + 3^y +5^y) /3 ]^(1/y)
=lim(y->0+) [ 1+ (1/3)ln(30)y ]^(1/y)
=e^[(1/3)ln(30)]
=(30)^(1/3)
=>
lim(n->∞) { [ 2^(1/n) + 3^(1/n) +5^(1/n) ]/3 }^n =(30)^(1/3)

追问

2^y = 1+(ln2)y +o(y)
3^y = 1+(ln3)y +o(y)
5^y = 1+(ln5)y +o(y)
这一部分不是很明白

追答

泰勒展式
f(y)= f(0) +f'(0)y + o(y)
f(y)= 2^y => f(0) =1
f'(y)= (ln2).2^y => f'(0) = ln2
f(y)= 1 +(ln2)y + o(y)