Limit of a function

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, the limit of a function is a fundamental concept in <a href="/facts/Calculus/MEmUTYoz">calculus</a> and <a href="/facts/Mathematical_analysis/XXeR26Ih">analysis</a> concerning the behavior of that <a href="/facts/Function_(mathematics)/CdReEKCh">function</a> near a particular <a href="/facts/Independent_variable/dINfzIF2">input</a> which may or may not be in the <a href="/facts/Domain_(mathematical_analysis)/xT44oI3E">domain</a> of the function.
Formal definitions, first devised in the early 19th century, are given below. Informally, a function f assigns an <a href="/facts/Dependent_variable/dINfzIF2">output</a> f(x) to every input x. We say that the function has a limit L at an input p, if f(x) gets closer and closer to L as x moves closer and closer to p. More specifically, the output value can be made arbitrarily close to L if the input to f is taken sufficiently close to p. On the other hand, if some inputs very close to p are taken to outputs that stay a fixed distance apart, then we say the limit does not exist.
The notion of a limit has many applications in <a href="/facts/Calculus/MEmUTYoz">modern calculus</a>. In particular, the many definitions of <a href="/facts/Continuous_function/sKbl02pB">continuity</a> employ the concept of limit: roughly, a function is continuous if all of its limits agree with the values of the function. The concept of limit also appears in the definition of the <a href="/facts/Derivative/7Ito70Dh">derivative</a>: in the calculus of one variable, this is the limiting value of the <a href="/facts/Slope/a2L3k0j5">slope</a> of <a href="/facts/Secant_line/gJnaF5g3">secant lines</a> to the graph of a function.

Limit of a function open-in-new

Limit of a function