Question

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

请给出详细的证明过程，O(∩_∩)O谢谢~

Answer 1

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+1<=3n，分母2(2n^2+1)>2(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

Answer 2

这些孩子都越来越狡猾了！或许是被逼出来的！