Question

如何用定义证明n的根号下n次方的极限是1

n趋于无穷
用ε-δ定义解答
谢谢

Answer 1

设An=n^(1/n)=1+Hn
n=(1+Hn)^n>n(n-1)*(Hn)^2/2

由上面的式子可知0<Hn<[2/(n-1)]^(1/2)
从而1<=An=1+Hn<=1+[2/(n-1)]^(1/2)

根据ε-δ定义，1+[2/(n-1)]^(1/2)的极限是1，所以由迫敛性得n^(1/n)=1

Answer 2

设An=n^(1/n)=1+Hn
n=(1+Hn)^n>n(n-1)*(Hn)^2/2
由上面的式子可知0<Hn<[2/(n-1)]^(1/2)
从而1<=An=1+Hn<=1+[2/(n-1)]^(1/2)
根据ε-δ定义，1+[2/(n-1)]^(1/2)的极限是1，所以由迫敛性得n^(1/n)=1

Answer 3

n^(1/n)

n---> wuqiong 1/n --->0

lim n^0 =1