Question

How do you find the limit of `(x^n-1) / (x-1)` as x approaches 1?

Answer 1

`lim_(x->1)(x^n-1)/(x-1)=n`

Explanation:

`lim_(x->1)(x^n-1)/(x-1) = lim_(x->1)((x-1)(x^(n-1)+x^(n-2)+...+x+1))/(x-1)`

`=lim_(x->1)(x^(n-1)+x^(n-2)+...+x+1)`

`=lim_(x->1)sum_(i=0)^(n-1)x^i`

`=sum_(i=0)^(n-1)1^i`

`=sum_(i=0)^(n-1)1`

`=n`