\( \definecolor{colordef}{RGB}{249,49,84} \definecolor{colorprop}{RGB}{18,102,241} \)
Going further. Let \(P \in \mathbb{C}[X]\) be a non-constant polynomial that splits as \(P = c \prod_i (X - a_i)^{m_i}\) with the \(a_i\) pairwise distinct and \(m_i \in \mathbb{N}^*\). Compute \(\displaystyle\lim_{X \to \infty} X \cdot \frac{P'(X)}{P(X)}\) from the partial-fraction decomposition of \(P' / P\), and check that the result agrees with \(\deg P\).
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