 Solution

Write $Q(x)=1-\Phi (x)={\int }_x^{\infty }\varphi (t)dt$, where $\varphi (t)=(2\pi {)}^{-1/2}{e}^{-t^2/2}$ is the standard normal density. For $x>0$, ${\varphi }^{\prime }(t)=-t\varphi (t)$ and integration by parts yieldThe boundary term at infinity vanishes. The remainder is nonnegative and is at most $Q(x)/x^2$, soDividing by the positive quantity $\varphi (x)/x$ gives lower and upper bounds $x^2/(x^2+1)$ and $1$. The squeeze theorem proves the Mills ratio asymptotic