Question

$A=\int_\lambda^\infty \frac{x}{\lambda}e^{-x^2/2}dx$
$B=\int_\lambda^\infty e^{-x^2/2}dx$
Eu quero encontrar:
$\boxed{\lim_{\lambda\to \infty}\frac AB}$
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Answer 1

Resposta:

$A(\lambda)=\frac1\lambda\int_{\lambda}^\infty x e^{-x^2/2} dx=\frac1\lambda e^{-\frac12\lambda^2}\\ A'(\lambda)=-\frac{1}{\lambda^2}e^{-\frac12\lambda^2}(\lambda^2+1) \\ B(\lambda)=\int_{\lambda}^\infty e^{-\frac12x^2}dx\\ B'(\lambda)=-e^{-\frac12\lambda^2} \\ \lim_{\lambda\to\infty}\frac{A(\lambda)}{B(\lambda)}=\lim_{\lambda\to\infty}\frac{A'(\lambda)}{B'(\lambda)}=\lim_{\lambda\to\infty}\frac{(\lambda^2+1)}{\lambda^2}\frac{e^{-\frac12\lambda^2}}{e^{-\frac12\lambda^2}}=1$