Limit of a function

The limit of a function is a fundamental concept in calculus and mathematical analysis describing how a function behaves near a particular input, whether or not that input actually lies in the function domain. Informally, a function f has a limit L at an input p if the value f of x gets arbitrarily close to L whenever x is taken sufficiently close to p, and if instead some inputs near p keep producing outputs that stay a fixed distance apart, the limit is said not to exist. Formal definitions of the concept were first developed in the early nineteenth century, and the notion underlies much of modern calculus: continuity is defined in terms of a function limits agreeing with its actual values, and the derivative is itself defined as the limiting value of the slope of secant lines to a function graph.