In Modeling Extremal Events by Embrechts et al [1997] page 130-131 it says that Taylor expansion can be used to prove: $ 1-\exp(-x^{-\alpha}) \sim x^{-\alpha}, \quad x \to \infty$ . But I don’t see how Taylor expansion would apply to this case. I proved it with L’Hôpital’s rule: $ \lim_{x \to \infty} \frac{1-\exp(-x^{-\alpha})}{x^{-\alpha}} = \lim_{xRead more