Limits tell calculus what a function is heading toward
In mathematics a limit is the value a function or sequence approaches as its input or index nears some point. That idea underwrites continuity, derivatives, and integrals across calculus and analysis, and later generalises to nets and category-theoretic limits.
A limit pins down where a function's outputs or a sequence's terms are heading as the input or index closes in on some point, and when an ordinary limit fails to exist, the limit inferior and limit superior still describe how the values are bounded. Hermann Hankel argued in 1871 that the modern concept descends from Proposition X.1 of Euclid's Elements, the result behind the method of exhaustion used by Euclid and Archimedes, which repeatedly strips away more than half of a magnitude.
The earliest definition of the limit of a geometric series, which Grégoire de Saint-Vincent named its terminus, appeared in his Opus Geometricum of 1647: an end the progression never reaches yet approaches more closely than any given segment. In the Scholium to the Principia of 1687, Isaac Newton treated ultimate ratios as limits that quantities approach until their difference falls below any stated amount.
Bernard Bolzano laid the groundwork for the epsilon-delta technique in 1817 while defining continuous functions, but other mathematicians did not learn of his work until thirty years after his death. It was Augustin-Louis Cauchy, in 1821, and later Karl Weierstrass who formalised what became the (ε, δ)-definition of the limit of a function. The familiar notation with an arrow written beneath 'lim' was devised by John Gaston Leathem in 1905 and caught on after G. H. Hardy used it in his textbook A Course of Pure Mathematics, published in 1908.
For sequences, a limit means the terms approximate a number arbitrarily well once finitely many early terms are thrown away: whatever error window is chosen, only a finite number of terms fall outside it. A sequence tends to infinity if all but finitely many terms exceed any given bound, as the positive integers do; such a sequence has an infinite limit but does not converge to a real number.
Source: Limit (mathematics)
