在数学中，函数极限是微积分学和数学分析的一个基本概念。它描述函数值在接近某一给定的自变量时的特征。

不严格地讲，函数 f 对于每个给定的在定义域内自变量，都会有一个对应的因变量 f(x)。声称在因变量为 p 时，函数极限为 L 则表明：当自变量 x 的值无限接近于 p 时，因变量 f(x) 的值便无限接近于 L。另一方面，如果存在十分接近于 p 的自变量所对应的因变量的值与 L 的值相差较大，则表示函数极限不存在。

极限的概念在现代微积分领域用途良多。比如，连续性的定义。除此之外，它还被用于导数的定义。

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. We say the function has a limit L at an input p: this means f(x) gets closer and closer to L as x moves closer and closer to p. More specifically, when f is applied to any input sufficiently close to p, the output value is forced arbitrarily close to L. On the other hand, if some inputs very close to p are taken to outputs that stay a fixed distance apart, we say the limit does not exist.