Limit of A 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.

Read more about Limit Of A Function:  History, Motivation, Definitions, Relationship To Continuity, Properties

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