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$$ \lim_{x \to \infty} \frac{\sigma - \sigma \Phi(x)}{(\mu + \sigma x)\phi(x)} $$ $$\stackrel{0/0}{=} \lim_{x \to \infty} \frac{-\sigma \phi(x)}{\sigma\phi(x) + (\mu + \sigma x)\phi'(x) } $$ $$ =\lim_{x \to \infty}\frac{-\sigma \phi(x)}{\sigma\phi(x) - (\mu + \sigma x)x\phi(x) } = 0$$ (Used $\phi'(x) = -x\phi(x) $ and $\Phi'(x) =\phi(x)$, just like in ...


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