Sequence limits ask what later terms are crowding toward
The limit of a sequence is the value its terms tend to, written with arrows or lim notation in analysis texts. Convergence means that beyond some index, every term stays arbitrarily close to that value—an idea that anchors infinite series and discrete approximations.
A sequence whose terms settle on a finite limit is convergent; one that does not is divergent, and a sequence tending to zero is sometimes called a null sequence. Formally, x is the limit if for every positive ε there is an index N beyond which every term lies within ε of x. The notion is often described as the foundation on which all of mathematical analysis rests, and although the idea works in every metric or topological space, most people first meet it with real numbers.
Zeno of Elea built paradoxes around limiting processes, and the method of exhaustion, credited to a line of thinkers from Leucippus and Democritus through Antiphon and Eudoxus to Archimedes, pins down areas and volumes through endless chains of approximations. In his Quadrature of the Parabola, Archimedes summed what is now called a geometric series to find the area trapped between a straight line and a parabolic arc. Grégoire de Saint-Vincent defined the terminus of a geometric series in 1647, and Pietro Mengoli's Geometriae speciosae elementa of 1659 used 'quasi-infinite' for unbounded and 'quasi-null' for vanishing quantities.
Newton wrote on series in works composed between 1669 and 1693, some published only decades later. In the 18th century Euler summed certain divergent series by stopping at a convenient point, caring less whether a limit existed than whether a number could be computed, and in 1797 Lagrange warned that this lack of rigour blocked further progress. Bolzano gave the modern definition in an 1816 Prague work on the binomial theorem that drew little notice, and Weierstrass supplied it again in the 1870s.
Any real number is the limit of its decimal approximations, so 0.333… is the limit of the partial sums of 3/10^k. Other limits are subtler: (1 + 1/n)^n tends to the number e, and the squeeze theorem often helps establish such results. Limits also respect arithmetic, so the limit of a sum, product, constant multiple or quotient of two convergent sequences equals the matching combination of their limits.
Source: Limit of a sequence
