Although the function (sin x)/x is not defined at zero, as x becomes closer and closer to zero, (sin x)/x becomes arbitrarily close to 1. We say that "the limit of (sin x)/x as x approaches zero equals 1."

Fundamental theorem Limits of functions Continuity Mean value theorem

Product rule Quotient rule Chain rule Change of variables Implicit differentiation Taylor's theorem Related rates Identities

Lists of integrals Improper integrals Integration by: parts, disks, cylindrical shells, substitution, trigonometric substitution, partial fractions, changing order

Gradient Divergence Curl Laplacian Gradient theorem Green's theorem Stokes' theorem Divergence theorem

Matrix calculus Partial derivative Multiple integral Line integral Surface integral Volume integral Jacobian

In mathematics, the limit of a function is a fundamental concept in calculus and analysis concerning the behavior of that function near a particular input. Informally, a function assigns an output f(x) to every input x. The function has a limit L at an input p if f(x) is "close" to L whenever x is "close" to p. In other words, f(x) becomes closer and closer to L as x moves closer and closer to p. More specifically, when f is applied to each input sufficiently close to p, the result is an output value that is arbitrarily close to L. If the inputs "close" to p are taken to values that are very different, the limit is said to not exist. Formal definitions, first devised in the early 19th century, are given below.