Smooth approximation near a non differentiable point

Let $$f:\mathbb{R}_+\rightarrow \mathbb{R}$$ be a function differentiable for $$x>0$$ but non differentiable at $$x=0$$ (for instance $$f=\sqrt{\cdot}$$) and $$g$$ be a polynomial function. I know how to compute the Taylor series of $$f(g(t,x))$$ at a point $$g(t_0,x_0)>0$$. But I don't know how to construct a differentiable approximation (polynomial) when this point is $$0$$.

Edit: an approximation of $$f$$ would be a smooth function $$\varphi_N$$ (polynomial) such that $$f(g(t,x)) = \varphi_N(t,x) + o(t^N,x^N)$$.

Are there some works in this direction ?

I read here How to Approximate at Non-differentiable Point (forced Series Expansion around Branch Cut) but it doesn't investigate this point.

• That depends very much on the meaning that you associate to the word "approximation". Dec 29, 2019 at 22:49
• Maybe a Pade approximation? Dec 30, 2019 at 0:10
• Maybe Needs["FunctionApproximations`"]; MiniMaxApproximation[Sqrt[h], {h, {$MachineEpsilon, 1}, 8, 2}] Dec 30, 2019 at 6:51 • I am having a look with Pade approximation, I can find a Padé approximation for$f$, does the composition with$g$will remain a approximation of$f\circ g$? Dec 30, 2019 at 9:08 • PadeApproximant[Sqrt[x],{x,0,{2,2}}] just returns Sqrt[x] and the reason for that is because Mathematica can't evaluate the Series of Sqrt[x] at$0\$ Dec 30, 2019 at 9:12