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In the book Asymptotic theory of statistics and probability by Anirban DasGupta (2008, Springer Science & Business Media) in page 109 Example 8.13 I found the following approximation

$$\Phi^{-1}\left(1-\frac{1}{n}\right)\approx \sqrt{2\log n}$$

and the authors said that it follows form

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

But i have been unable to derive any of them. Can anyone give me some insite on the proofs or point to a source where these kind of approximations are discussed in more detail?

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