# Limit of a function

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

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,

although the function

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.

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.