Calculus
Calculus was originally motivated by two classical problems: finding the slope of the tangent line to a curve at a point, and finding the area underneath the curve. What's 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, and is in fact 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.
