Question

求极限的题目

Answer 1

取自然对数，
指数＝5nln[(1＋2¹/ⁿ＋3¹/ⁿ) / 3]
＝5nln[1＋(2¹/ⁿ - 1)/3＋(3¹/ⁿ - 1)/3]
≈ 5n[(2¹/ⁿ -1)/3＋(3¹/ⁿ - 1)/3]
由于 a^x＝e^(xlna) ≈ 1＋xlna(x→0)，
所以指数 ≈ 5n[ln2/(3n)＋ln3/(3n)]
＝5ln6 / 3＝ln[6^(5/3)]，
所以原极限＝e^ln[6^(5/3)]＝6^(5/3) 。

Answer 2

如图所示

Answer 3

lim(n->∞) { [ 1+ 2^(1/n)+ 3^(1/n) ] /3 }^(5n)
consider
lim(x->∞) { [ 1+ 2^(1/x)+ 3^(1/x) ] /3 }^(5x)
let
y=1/x
y->0
2^y = 1+(ln2)y +o(y)
3^y = 1+(ln3)y +o(y)
(1+2^y +3^y)/3 = 1 + (1/3)(ln6)y +o(y)
lim(x->∞) { [ 1+ 2^(1/x)+ 3^(1/x) ] /3 }^(5x)
=lim(y->0) [ ( 1+ 2^y+ 3^y ) /3 ]^(5/y)
=lim(y->0) [ 1 +(1/3)(ln6)y ]^(5/y)
=e^[ (5/3)ln6 ]
=6^(5/3)
=>
lim(n->∞) { [ 1+ 2^(1/n)+ 3^(1/n) ] /3 }^(5n) = 6^(5/3)