A limit describes the value that a function approaches as the input approaches a particular value.
The limit of f(x) as x approaches a is written as limx→a f(x). Limits are fundamental to calculus, providing the rigorous foundation for derivatives and integrals. A limit exists if the function approaches the same value from both sides (left-hand and right-hand limits). Limits can be finite, infinite, or may not exist. They are used to analyze continuous and discontinuous functions.
The concept of limits was intuitively used by Newton and Leibniz in developing calculus, but lacked rigorous definition. Augustin-Louis Cauchy provided the first formal definition in the 1820s. Karl Weierstrass later refined it into the epsilon-delta (ε-δ) definition still used today. Limits resolved paradoxes like Zeno's paradoxes and made calculus mathematically sound.