Lim [x->1] {x^(1/3) - 1} / {x^(1/4) - 1}
Let x = u^12, then the limit becomes,
Lim [u->1] {u^4 -1} / {u^3 - 1}
(u^4 - 1) can be writtten as (u^2 - 1)(u^2 + 1) = (u - 1)(u + 1)(u^2 + 1)
(u^3 - 1) can be written as (u - 1)(u^2 + u + 1)
The limit now becomes,
Lim [u -> 1] (u - 1)(u + 1)(u^2 + 1) / (u - 1)(u^2 + u + 1)
Lim [u -> 1] (u + 1)(u^2 + 1) / (u^2 + u + 1)
As u -> 1, Lim -> 2*2 / 3 = 4/3
Limit = 4/3