Limit (mathematics)

In formulas, a limit of a function is usually written as


and is read as "the limit of f of x as x approaches c equals L". This means that the value of the function f can be made arbitrarily close to L, by choosing x sufficiently close to c. Alternatively, the fact that a function f approaches the limit L as x approaches c is sometimes denoted by a right arrow (→ or ${\displaystyle \rightarrow }$), as in


which reads "${\displaystyle f}$ of ${\displaystyle x}$ tends to ${\displaystyle L}$ as ${\displaystyle x}$ tends to ${\displaystyle c}$".


The concept of a limit of a sequence is further generalized to the concept of a limit of a topological net, and is closely related to limit and direct limit in category theory.


The limit inferior and limit superior provide generalizations of the concept of a limit which are particularly relevant when the limit at a point may not exist.

History[edit]

According to Hankel (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: "Two unequal magnitudes being set out, if from the greater there is subtracted a magnitude greater than its half, and from that which is left a magnitude greater than its half, and if this process is repeated continually, then there will be left some magnitude less than the lesser magnitude set out."^[2][3]


Grégoire de Saint-Vincent gave the first definition of limit (terminus) of a geometric series in his work Opus Geometricum (1647): "The terminus of a progression is the end of the series, which none progression can reach, even not if she is continued in infinity, but which she can approach nearer than a given segment."^[4]


The modern definition of a limit goes back to Bernard Bolzano who, in 1817, developed the basics of the epsilon-delta technique to define continuous functions. However, his work remained unknown to other mathematicians until thirty years after his death.^[5]


Augustin-Louis Cauchy in 1821,^[6] followed by Karl Weierstrass, formalized the definition of the limit of a function which became known as the (ε, δ)-definition of limit.


The modern notation of placing the arrow below the limit symbol is due to G. H. Hardy, who introduced it in his book A Course of Pure Mathematics in 1908.^[7]