Question

How to solve `lim(root(n)(x)-1)/(root(m)(x)-1)` with `n,m in NN` and `n>=2`?

Answer 1

`m/n`

Explanation:

Choosing now `y^(nm) = x` and substituting

`(root(n)(x)-1)/(root(m)(x)-1)equiv (y^m-1)/(y^n-1) = (1+y+cdots+y^(m-1))/(1+y+cdots+y^(n-1))`

and finally

`lim_(x->1)(root(n)(x)-1)/(root(m)(x)-1)=lim_(y->1)(1+y+cdots+y^(m-1))/(1+y+cdots+y^(n-1)) = m/n`

NOTE

We were using the polynomial identity

`(x^n-1)/(x-1) = 1+x+x^2+ cdots+ x^(n-1)`