Numerical analysis is the art of applying theoretical mathematics to scientific problems.
Numerical analysis shows how to estimate these limits with finite calculations, and numerical analysis determins the speed and accuracy of the approximation algorithms. Numerical analysis was invented before computers, but computing has vastly increased the importance of numerical analysis. Before computers, a practical algorithm had to reduce a calculation to at most a few thousand steps. Very few significant algorithms could be developed that were so short. With computers, an algorithm is practical if it reduces a calculation to only millions of steps.