congratulations, you have implmented the Zariski tangent space using nilpotent matrices. welcome to the beautiful theory of algebraic geometry and schemes.

http://math.stanford.edu/~vakil/0708-216/216class21.pdf

This really does fall in the ream of algebraic geometry since this method only works for rational functions - as he implemented it.

To numerically compute sin(x + ε) you need the Taylor series.