Johnstone and Silverman (2005) claimed that for large x 


$\frac{1-\Phi(x)}{\phi(x)} \approx \frac{1}{x}$ 


where $\Phi(x)$ and $\phi(x)$ are the CDF and PDF for a normal random variable. 


I was able to verify the claim numerically. Q: But how would I show this analytically? This seems like it should be easy, but I can't figure it out. Also, Q: Is there a symbolic logic system (e.g., Mathematica) that can generate these sort of approximations? 



Approximation to the ratio of a Gaussian CDF to PDF
If you interpret this as the existence of the limit
$$
\lim\_{x \rightarrow \infty} \frac{x(1-\Phi(x))}{\phi(x)}
$$
then it is easy to verify using l'Hopital's rule.


