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I dont get why $$\lim_{x \to \infty} \frac{\mu \{1 - \Phi(x)\} + \sigma \phi(x)}{(\mu + \sigma x) \{1 - \Phi(x)\} } = \lim_{x \to \infty} \frac{1}{1 - \sigma \frac{1 - \Phi(x)}{(\mu + \sigma x)\phi(x)}}, $$. This is the only part that I'm stuck with, otherwise I don't see how to evaluate the limit. Thanks for reading

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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 your first L'Hospital application to get your equality. And L'Hospital works the second time too as $\lim_{x \to \infty}\phi(x) = 0$, $\lim_{x \to \infty}x \phi(x) = 0$.)

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