Calculus
The discovery of the branch of mathematics known as calculus was motivated by two classical problems: how to find the slope of the tangent line to a curve at a point and how to find the area bounded by a curve. What is surprising is that these two problems are fundamentally connected and, together with the notion of limits, can be used to analyse instantaneous rates of change, accumulations of change, volumes of irregular solids, and much more.
Limits are heavily used in calculus. The formal notion of a limit is what "differentiates" (hehe, pun) calculus from precalculus mathematics.
The use of calculus in pre-collegiate mathematics competitions is generally frowned upon. However, many physics competitions require it, as does the William Lowell Putnam competition.
The subject dealing with the rigorous foundations of calculus is called analysis, specifically real analysis.
