Editing Limit of a function (section)

{{Short description|Point to which functions converge in analysis}}
{{For|the mathematical concept in general|Limit (mathematics)}}
{{Use dmy dates|date=April 2020}}
<div class="thumb tright">
<div class="thumbinner" style="width:252px;">
{| class="wikitable" style="width:100%; margin:0px;"
!<math>x</math>!!<math>\frac{\sin x}{x}</math>
|-
|1||0.841471...
|-
|0.1||0.998334...
|-
|0.01||0.999983...
|}
<div class="thumbcaption">
Although the function {{tmath|\tfrac{\sin x}{x} }} is not defined at zero, as {{mvar|x}} becomes closer and closer to zero, {{tmath|\tfrac{\sin x}{x} }} becomes arbitrarily close to 1. In other words, the limit of {{tmath|\tfrac{\sin x}{x},}} as {{mvar|x}} approaches zero, equals&nbsp;1.
</div>
</div>
</div>

{{Calculus}}
In [[mathematics]], the '''limit of a function''' is a fundamental concept in [[calculus]] and [[mathematical analysis|analysis]] concerning the behavior of that [[Function (mathematics)|function]] near a particular [[independent variable|input]] which may or may not be in the [[Domain (mathematical analysis)|domain]] of the function.

Formal definitions, first devised in the early 19th century, are given below. Informally, a function {{mvar|f}} assigns an [[dependent variable|output]] {{math|''f''(''x'')}} to every input {{mvar|x}}. We say that the function has a limit {{mvar|L}} at an input {{mvar|p}}, if {{math|''f''(''x'')}} gets closer and closer to {{mvar|L}} as {{mvar|x}} moves closer and closer to {{mvar|p}}. More specifically, the output value can be made ''arbitrarily'' close to {{mvar|L}} if the input to {{mvar|f}} is taken ''sufficiently'' close to {{mvar|p}}. On the other hand, if some inputs very close to {{mvar|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 [[Calculus#Modern|modern calculus]]. In particular, the many definitions of [[continuous function|continuity]] 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 [[derivative]]: in the calculus of one variable, this is the limiting value of the [[slope]] of [[secant line]]s to the graph of a function.