Limit of a function

From Knowino
Jump to: navigation, search
The function  tends towards e as 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

 \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 be defined at values of the argument at which the function itself is not defined. For example,

 \lim_{x \to 0}\frac{\sin(x)}{x} = 1 ,

although the function

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

is not defined at x=0.

Formal definition

Let f be a function defined (at least) on some open interval containing a (except possibly at a) and let L be a real number. Then the equality

 \lim_{x \to a}f(x) = L

means that

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

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

See also

Information.svg Some content on this page may previously have appeared on Citizendium.
Personal tools
Variants
Actions
Navigation
Community
Toolbox