Limit of a function: Difference between revisions

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{{Image|Plot-e.png|right|300px|The function <math>(1 + \tfrac{1}{n})^n</math> tends towards <math>e</math> as <math>x</math> tends towards infinity.}}
In [[mathematics]], the concept of a '''limit''' is used to describe the behavior of a [[function (mathematics)|function]] as its [[argument]] either "gets close" to some point, or as it becomes arbitrarily large.
In [[mathematics]], the concept of a '''limit''' is used to describe the behavior of a [[function (mathematics)|function]] as its [[argument]] either "gets close" to some point, or as it becomes arbitrarily large.



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The function tends towards as 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

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".

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.

See also