Question

lim(n趋向于正无穷)(x^n-1)/(x^n+1)=?,求详细过程，谢谢大家！

Answer 1

1. |x|>1,即x<-1或x>1
lim(n趋向于正无穷)(x^n-1)/(x^n+1)
=lim(n趋向于正无穷)(1-1/x^n)/(1-1/x^n)
=1
2.-1<x<1
lim(n趋向于正无穷)(x^n-1)/(x^n+1)
=lim(n趋向于正无穷)(0-1)/(0+1)
=-1
3.x=-1
无意义!
4.x=1
lim(n趋向于正无穷)(x^n-1)/(x^n+1)=0

Answer 2

当x<-1或x>1，原式=lim(n趋向于正无穷)[1-(1/x)^n]/[1+(1/x)^n]=(1-0)/(1+0)=1
当x=-1，原式=lim(n趋向于正无穷)[(-1)^n-1]/[(-1)^n+1] 极限不存在
当-1<x<1，原式=(0-1)/(0+1)=-1
当x=1，原式=(1-1)/(1+1)=0

Answer 3

lim(n->无穷){(x^n-1)/(x^n+1)}
=lim(n->无穷){1-2/(x^n+1)}
if |x|<1
lim(n->无穷){1-2/(x^n+1)}
=1-2 =-1

if |x|>=1
lim(n->无穷){1-2/(x^n+1)}
=1-0 =1