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Let \(f\) be the function defined on \(\mathbb{R}\) by \(f(x) = \ln(e^{-x} + 1)\), and let \(\mathscr{C}\) be its graphical representation.
  1. Justify that \(f\) is well-defined on \(\mathbb{R}\).
    1. Study the limits of \(f\) at \(-\infty\) and \(+\infty\).
    2. Deduce the existence of a horizontal asymptote.
    1. Determine the derivative function \(f'\).
    2. Deduce the variations of \(f\) on \(\mathbb{R}\).
    1. Show that for all \(x \in \mathbb{R}\), \(f(x) = -x + \ln(1 + e^x)\).
    2. Let \(d\) be the line with equation \(y = -x\). Determine the limit of \(f(x) + x\) as \(x \to -\infty\) and provide a graphical interpretation.
    3. Study the relative position of \(\mathscr{C}\) and \(d\).

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