# Limit of a function

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The function ${\displaystyle (1+{\tfrac {1}{n}})^{n}}$ tends towards ${\displaystyle e}$ as ${\displaystyle n}$ tends towards infinity.

In mathematics, the concept of a limit is used to describe the behavior of a function as its argument either "gets close" to some point, or as it becomes arbitrarily large.

Suppose f(x) is a real-valued function and a is a real number. The expression

${\displaystyle \lim _{x\to a}f(x)=L}$

means that f(x) can be made arbitrarily close to L by making x sufficiently close to a. We say that "the limit of the function f of x, as x approaches a, is L". This does not necessarily mean that f(a) is equal to L, or that the function is even defined at the point a.

Limit of a function can in some cases be defined even at values of the argument at which the function itself is not defined. For example,

${\displaystyle \lim _{x\to 0}{\frac {\sin(x)}{x}}=1,}$

although the function

${\displaystyle f(x)={\frac {\sin(x)}{x}}}$

is not defined at x=0.

### Formal definition

Let f be a function defined on an open interval containing a (except possibly at a) and let L be a real number.

${\displaystyle \lim _{x\to a}f(x)=L}$

means that

for each real ε > 0 there exists a real δ > 0 such that for all x with 0 < |x − a| < δ, we have |f(x) − L| < ε.

This formal definition of function limit is due to the German mathematician Karl Weierstrass.