Approximating Error in $\partial_x u(x;\eta)$

[bruno67]

I have a quantity $U(x)$, x>0, which I cannot calculate exactly. Numerically, I can calculate an approximation $u(x;\eta)$, for $\eta>0$, which is very close to $U(x)$ if $\eta$ is small enough. I know that the error $\xi(x;\eta)=u(x;\eta)-U(x)$ satisfies an estimate
$$|\xi(x;\eta)|\le E(x;\eta)$$
where $\lim_{\eta\to 0}E(x;\eta)=0$ for all x, and I can use this to choose my parameter $\eta$ so that the error lies under a specified tolerance.

Based only on the above, is it possible to derive an approximate estimate for the error in $\partial_x u(x;\eta)$, i.e. $\partial_x \xi(x;\eta)$?

 

[fresh_42]

No. As far as I can tell from the description, we have good information for any specific value of $x$ but none about the dependence of $x$. And even if everything tends to zero, there could still be high slopes locally.