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The concept of limit has multiple meanings. It can be a line separating two territories, from one extreme reached after awhile or a restriction (or limitation).
In mathematics, a limit is a fixed size that approximates to the terms of a suite/infinite sequence of quantities.
Function, on the other hand, is a concept that refers to various questions. In this case, we will discuss a mathematical function (the f report of the elements of a set A with the elements of a set B).
The term limit of a function is used in the mathematical calculus and refers to the proximity between a value and a point. For example, if a function f has a limit X on a point t, this means that the value of f can be anything that is close to X as desired, with points sufficiently close to t but different.
The limits of functions were already analysed in the 17th century, although the modern notation was born in the 18th century on the basis of the work of various specialists. It is said that Karl Weierstrass was the first mathematician to have proposed a precise technique, between 1850 and 1860.
In short, a function f with a X limit on t means that this function tends to its X limit near t, with f(x) as close to X as possible, but so that x is different from t. However, the idea of proximity is not very precise, reason for which a formal definition requires more elements.