Question

`lim_(n -> oo) n^(1/n)=?` for `n in NN` ?

Given `x_n={""^nsqrt(n)}_(n=1)^oo`
thus `x_1=1,x_2=sqrt2,x_3=""^3sqrt(3),...`
prove: `lim_(n -> oo) x_n=1`

Answer 1

1

Explanation:

`f(n)=n^(1/n) implies log(f(n))=1/n log n`

Now

`lim_{n -> oo}log(f(n)) = lim_{n -> oo} log n/n`
`qquadqquadqquad = lim_{n -> oo} {d/(dn) log n}/{d/(dn) n} = lim_{n-> oo}(1/n)/1=0`

Since `log x` is a continuous function, we have

`log (lim_{n to oo}f(n))=lim_{n to oo} log(f(n)) = 0 implies`

`lim_{n to oo}f(n)=e^0=1`