Limit of A Function | Limit Function

Limit Of A Function

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. In other words, the limit of (sin x)/x as x approaches zero equals 1.

Topics in calculus
Fundamental theorem Limits of functions Continuity Mean value theorem Rolle's theorem
Differential calculus Derivative Second derivative Third derivative Change of variables Implicit differentiation Taylor's theorem Related rates Rules and identities Power rule Product rule Quotient rule Sum rule Chain rule
Integral calculus Lists of integrals Improper integral Multiple integral Integration by parts disks cylindrical shells substitution trigonometric substitution partial fractions changing order
Vector calculus Gradient Divergence Curl Laplacian Gradient theorem Green's theorem Stokes' theorem Divergence theorem
Multivariable calculus 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.

Formal definitions, first devised in the early 19th century, are given below. Informally, a function f 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.

The notion of a limit has many applications in modern calculus. In particular, the many definitions of continuity employ the limit: roughly, a function is continuous if all of its limits agree with the values of the function. It also appears in the definition of the derivative: in the calculus of one variable, this is the limiting value of the slope of secant lines to the graph of a function.

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