(ε, δ)-definition of limit

In calculus, the (ε, δ)-definition of limit ("epsilon–delta definition of limit") is a formalization of the notion of limit. The concept is due to Augustin-Louis Cauchy, who never gave an (${\displaystyle \varepsilon ,\delta }$) definition of limit in his Cours d'Analyse, but occasionally used ${\displaystyle \varepsilon ,\delta }$ arguments in proofs. It was first given as a formal definition by Bernard Bolzano in 1817, and the definitive modern statement was ultimately provided by Karl Weierstrass. It makes rigorous the following informal notion: the dependent expression f(x) approaches the value L as the variable x approaches the value c if f(x) can be made as close as desired to L by taking x sufficiently close to c.

Although the Greeks examined limiting process, such as the Babylonian method, they probably had no concept similar to the modern limit. The need for the concept of a limit came into force in the 17th century when Pierre de Fermat attempted to find the slope of the tangent line at a point ${\displaystyle x}$ of a function such as ${\displaystyle f(x)=x^{2}}$. Using a non-zero, but almost zero quantity, ${\displaystyle E}$, Fermat performed the following calculation:

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