information:extraction, structural:scholar_edition
History

Although implicit in the development of calculus of the 17th and 18th centuries, the modern idea of the limit of a function goes back to Bolzano who, in 1817, introduced the basics of the epsilon-delta technique (see (ε, δ)-definition of limit below) to define continuous functions. However, his work was not known during his lifetime.

In his 1821 book Cours d'analyse, Augustin-Louis Cauchy discussed variable quantities, infinitesimals and limits, and defined continuity of y = f (x) {\displaystyle y=f(x)} by saying that an infinitesimal change in x necessarily produces an infinitesimal change in y, while Grabiner claims that he used a rigorous epsilon-delta definition in proofs. In 1861, Weierstrass first introduced the epsilon-delta definition of limit in the form it is usually written today. He also introduced the notations lim {\textstyle \lim } and lim x → x 0. {\textstyle \textstyle \lim _{x\to x_{0}}\displaystyle.}

The modern notation of placing the arrow below the limit symbol is due to Hardy, which is introduced in his book A Course of Pure Mathematics in 1908.