f(1)=0,f'(1)=2,求极限limn→∞[nf(n/n+2）

Question

如题所述

Answer 1

lim(n->∞) n f(n/(n+2))
consider
lim(x->∞) x f(x/(x+2))
=lim(x->∞) f(x/(x+2)) / (1/x) (0/0 分子分母分别求导)
=lim(x->∞) [ 2/(x+2)^2] .f'(x/(x+2)) / (-1/x^2)
=lim(x->∞) [ -x^2/(x+2)^2] .f'(x/(x+2))
=lim(x->∞) [ -x^2/(x+2)^2] . lim(x->∞) f'(x/(x+2))
=-f'(1)
=-2
=>
lim(n->∞) n f(n/(n+2)) =-2

Answer 2

详细过程如图所示，希望能帮到你

追答

不好意思，应该这样做

看这个过程

差点忽略一个问题

修改下