Question

How do you evaluate the limit `(root3x-1)/(sqrtx-1)` as x approaches `1`?

Answer 1

`2/3`

Explanation:

`(root3x-1)/(sqrtx-1) equiv (y^2-1)/(y^3-1)` by doing the substitution #y = root6 x#

but

`(y^2-1)/(y^3-1) =( (y+1)(y-1))/(y^3-1) = (y+1)/(y^2+y+1)`

so

`lim_{x->1}(root3x-1)/(sqrtx-1) equiv lim_{y->1} (y+1)/(y^2+y+1) = 2/3`