Question

数学证明题目：用数列极限证明lim(n^2+n+1)/(2n^2+1)=1/2

Answer 1

因为|xn-a|=(n+3)/(n^2+n+1)≤4n/n^2=4/n，所以对于任意小的正数ε，要使得|xn-a|＜ε。只要4/n＜ε，即n＞4/ε。取正整数n=[4/ε]，n＞n时，恒有|xn-a|＝|(n^2-2)/(n^2+n+1)-1|＜ε。
所以，lim(n^2-2)/(n^2+n+1)=1

Answer 2

对于任意ε>0
令N=max(1,3/(4ε))
当n>N时
|(n^2+n+1)/(2n^2+1)-1/2|
=|2n^2+2n+2-2n^2-1|/[2(2n^2+1)]
=(2n+1)/[2(2n^2+1)]
分子2n+12(2n^2)=4n^2
<3n/[4n^2]
=3/4n<3/4(3/4ε)=ε
所以由极限定义
lim(n^2+n+1)/(2n^2+1)=1/2