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Let \(g\) be the function defined on \(D_g = (0, e) \cup (e, +\infty)\) by:$$ g(x) = \dfrac{\ln(x) + 1}{\ln(x) - 1} $$
  1. Show that for all \(x \in D_g\), \(g'(x) = \dfrac{-2}{x(\ln(x)-1)^2}\).
  2. Determine the variations of \(g\) on its domain of definition.

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