How would one usefully evaluate the function $f(x)=\frac{1-\cos (x)}{x}$ for arguments $0<x \ll 1$ evaluate? Calculate $f\left(10^{-4}\right)$ with an error smaller than $10^{-10}$.
Unfortunately, I have not yet had much to do with the field of numerics. What is clear to me is this: $$\kappa_{f}(x)=\left|\frac{x f^{\prime}(x)}{f(x)}\right|=\frac{\left(\frac{1-\cos x}{x}\right)^{\prime} \cdot x}{\frac{1-\cos x}{x}}$$ However, I do not know how to interpret this for the given area $0<x \ll 1$ and how I then carry out this error calculation.