Setup: Let us denote the pdf and cdf functions of the standard Gaussian distribution by $\phi(x)$ and $\Phi(x)$ respectively. What are the asymptotic scalings of both of them (in terms of their inputs)? I can get the pdf's scaling but not the cdf's scaling.
My attempt: For the pdf, we have \begin{align*} \phi(x) =\frac{\exp(-x^2/2)}{\sqrt{2\pi}} =\Theta(\exp(-x^2/2)). \end{align*} So for example, if $x=\Theta((\log n)^{1/2})$, then $\phi(x)=n^{-1/2}$.
Question: What about the cdf? The definition of the cdf has an integration, making it hard to simplify. If the big-$\Theta$ scaling is not possible, then any tight upper bound (big-$O$ scaling) also works. Thanks. I would prefer something simple that avoids the use of the erf functions.
Related work: This answer provided something simple for $x>0$, I am looking for something similar but for general $x$ or $x\leq0$.