Short answers, pulled from the story.
Archimedes used the method of exhaustion to compute the area inside a circle by finding the area of regular polygons with more and more sides. The Greek mathematician Eudoxus also made explicit use of limits when he applied the same technique to calculate volumes.
Real analysis began to emerge as an independent subject when Bernard Bolzano introduced the modern definition of continuity in 1816. Cauchy began to put calculus on a firm logical foundation by rejecting the principle of the generality of algebra widely used in earlier work in 1821.
Descartes's publication of La Géométrie in 1637 introduced the Cartesian coordinate system and is considered to be the establishment of mathematical analysis. This event marked the beginning of analytic geometry which allowed mathematicians to determine the maxima and minima of functions.
Archimedes' work The Method of Mechanical Theorems rediscovered in the 20th century showed his explicit use of infinitesimals. These ancient methods laid the groundwork for later formal theories without yet having a rigorous definition of continuity.
Modern numerical analysis does not seek exact answers because exact answers are often impossible to obtain in practice. Much of numerical analysis is concerned with obtaining approximate solutions while maintaining reasonable bounds on errors.