Infinite values are approximations to large finite values but are easier to work with: "Vladimir Arnold [3] forcefully stated that it is wrong to think about f

Infinite values are approximations to large finite values but are easier to work with: "Vladimir Arnold [3] forcefully stated that it is wrong to think about finite difference equations as approximations of differential equations. It is the differential equation which approximates finite difference laws of physics; it is the result of taking an asymptotic limit at zero. Being an approximation, it is easier to solve and study. In support to his thesis, Arnold refers to a scene almost everyone has seen: old tires hanging on sea piers to protect boats from bumps. If you control a boat by measuring its speed and distance from the pier and select the acceleration of the boat as a continuous function of the speed and distance, you can come to the complete stop precisely at the wall of the pier, but only after infinite time: this is an immediate consequence of the uniqueness theorem for solutions of differential equations. To complete the task in sensible time, you have to allow your boat to gently bump into the pier. The asymptotic at zero is not always an ideal solution in the real world. But it is easier to analyze!" From page 141 of Mathematics under the Microscope by Alexandre Borovik: http://eprints.maths.manchester.ac.uk/844/1/MicroMathWeb.pdf http://eprints.maths.manchester.ac.uk/844/1/MicroMathWeb.pdf