Bolzano and Cauchy turned fuzzy approach into formal limits
The limit of a function describes outputs near an input that may itself lie outside the domain. Informal talk says f(x) nears L as x nears a; early-nineteenth-century definitions by Bolzano and Cauchy made that talk proof-ready after centuries of calculus practice.
The limit of a function describes what outputs do near an input, and that input need not lie in the domain at all. Continuity is defined through limits, a function being continuous roughly when its limits agree with its values, and the derivative is itself defined as a limit. Augustin-Louis Cauchy's Cours d'analyse of 1821 described continuity by saying an infinitesimal change in x must produce an infinitesimal change in y, though the historian Judith Grabiner argues he used rigorous epsilon-delta reasoning in his proofs.
Karl Weierstrass introduced the epsilon-delta definition in its now-standard form in 1861, and G. H. Hardy's A Course of Pure Mathematics of 1908 popularised the arrow under the limit symbol. The definition says f(x) has limit L at p if for every positive ε some positive δ exists so that whenever x lies within δ of p, but is not p itself, f(x) lies within ε of L. Cauchy sometimes used ε as shorthand for 'error', and the pair can be read as error and distance.
A worked example: 4x + 1 tends to 9 as x approaches 2, because choosing δ = ε/4 keeps the output within ε of 9. The value at p itself is irrelevant: (2x² − x − 1)/(x − 1) is undefined at 1, yet its limit there is 3, since δ = ε/2 does the job. For a function defined on [0, 1) together with (1, 2], the definition lets a limit exist at 1 but not at the endpoints 0 or 2.
The input may also approach from only one side, giving right-hand and left-hand limits. If both exist and agree, their common value is the limit; if they exist but differ, or if either fails to exist, the function has no limit at that point.
Source: Limit of a function
