数列 极限 证明证明：若(1)y(n+1)>y(n)(2)lim yn->∞(3)lim(x(n+1)-x(n))/(y(n+1)-yn)存在那么lim xn/yn=lim(x(n+1)-x(n))/(y(n+1)-yn)
问题描述：
数列 极限 证明
证明：若
(1)y(n+1)>y(n)
(2)lim yn->∞
(3)lim(x(n+1)-x(n))/(y(n+1)-yn)存在
那么lim xn/yn=lim(x(n+1)-x(n))/(y(n+1)-yn)
答
还是老样子,极限的定义,无限分有限+无限
lim(x(n+1)-x(n))/(y(n+1)-yn)存在
设lim(x(n+1)-x(n))/(y(n+1)-yn)=a
对于任意e>0,存在N使得,对n>N有
|(x(n+1)-x(n))/(y(n+1)-yn)-a|
a-e(a-e)(y(n+1)-yn)
(a-e)(y(N+2)-y(N+1))
(a-e)(y(n+1)-yn)
(a-e)(y(n+1)-y(N+1))
|(x(n+1)-x(N+1))/(y(n+1)-y(N+1))-a|
xn/yn - a=(xn-x(N+1))/(yn-y(N+1))* (yn-y(N+1))/yn+(x(N+1)-a*y(N+1))/yn 凑出上式
|xn/yn - a|存在N'>N使得对n>N'有
|(x(N+1)-a*y(N+1))/yn|
有|xn/yn - a|那么lim xn/yn=lim(x(n+1)-x(n))/(y(n+1)-yn)