Short answers, pulled from the story.
A limit is the value that a function or sequence approaches as its argument or index approaches some value. Limits are essential to calculus and mathematical analysis, where they are used to define continuity, derivatives, and integrals.
Bernard Bolzano developed the basics of the epsilon-delta technique in 1817 to define continuous functions, though his work stayed unknown for thirty years after his death. Augustin-Louis Cauchy formalized the limit of a function in 1821, and Karl Weierstrass followed, producing the (ε, δ)-definition of limit.
According to Hankel in 1871, the modern concept of limit originates from Proposition X.1 of Euclid's Elements, which forms the basis of the Method of exhaustion found in Euclid and Archimedes. Grégoire de Saint-Vincent gave the first definition of a limit in his 1647 work Opus Geometricum.
The notation of placing the arrow below the limit symbol was invented by John Gaston Leathem in 1905. It was popularized by G. H. Hardy's 1908 textbook A Course of Pure Mathematics.
A Cauchy sequence is one whose terms eventually all lie within an interval of any arbitrarily small diameter, a property depending on the sequence alone rather than its limit. For real numbers, every Cauchy sequence is convergent, but in a general metric space the converse can fail. A space where every Cauchy sequence converges is called a complete metric space.
A limit fails to exist when the left-handed limit from below and the right-handed limit from above disagree. The positive indicator function illustrates this, having a left-handed limit of 0 and a right-handed limit of 1 at zero, so its limit there does not exist.