Evaluate the trend of two infinities?

[jc2411 (Jim Choi)]

[gianlucagorni]

Integrate[1/(x - 1), {x, 0, 3}, PrincipalValue → True]

[mariuszi28cg (Mariusz Iwaniuk)]

We have singularity at point “1”.Then from wikipedia:

int = Integrate[1/(x - 1), {x, 0, 1 - e}, Assumptions -> e >= 0] + 
      Integrate[1/(x - 1), {x, 1 + e, 3}, Assumptions -> e >= 0]
Limit[int[[1]], e -> 0, Direction -> "FromAbove"]

(* Log[2] *)