Is this double limit for logarithms true?

Question

Mathematica knows that: $$\gamma = \lim_{n\to \infty } \, \lim_{s\to 0} \, \left(\int \frac{(s+1)^{-\exp (n)-1}+s-1}{s} \, ds+\frac{(s+1)^{-n-1}+s-1}{s}\right)$$

Where $\gamma$ is Euler Gamma ($\gamma=0.57721566\dots$)

The question is if this limit is true:

$$\log (x)=\lim_{n\to \infty } \, \lim_{s\to x} \, \left(-\left(\frac{(s+1)^{-n-1}+s-1}{s}+\int \frac{(s+1)^{-e^n-1}+s-1}{s} \, ds\right)+\frac{s-1}{s}+s\right)$$

Mathematica:

Clear[n, x, s]
x = N[3];
Limit[Limit[(s + (s - 1)/
      s) - (Integrate[((s + 1)^(-Exp[n] - 1) + s - 1)/s, 
      s] + ((s + 1)^(-n - 1) + s - 1)/s ), s -> x], n -> 1000]