Let $\mu$ be a function from $\mathbb R_+\rightarrow\mathbb R_+$ in $C^\infty$ with $\Upsilon:=\sup\{|p|:\mu(|p|^2)>0\}$. Suppose that $$\lim_{|p|\rightarrow \Upsilon}\frac{\mu(|p|^2)}{(\Upsilon-|p|)^n}$$ exists and is positive.
Define $$\varphi(u):=\int_{\mathbb R^{d-1}}\mu(u^2+|w|^2)\,dw.$$ I am trying to prove that $$\lim_{|u|\rightarrow \Upsilon}\frac{\varphi(u)}{(\Upsilon-|u|)^{n+{\frac{d-1}{2}}}}$$ exists and is positive.
My only idea is to change to polar coordinates:
In polar coordinates, $$\varphi(u):=c_{d-1}\int_{0}^{\sqrt{\Upsilon^2-|u|^2}}\mu(u^2+r^2)r^{d-2}\,dr$$
and then use the fact that $\mu(|\cdot|^2)$ close to $\Upsilon$ behaves like $\mu(|u|^2)\sim (\Upsilon-|u|)^n$. Therefore close to $\Upsilon$,$$\varphi(u)\sim\int_0^{\sqrt{\Upsilon^2-|u|^2}}(\Upsilon-(|u|^2+r^2)^{\frac{1}{2}})^nr^{d-2}\,dr.$$ But I'm not sure where to go from here.