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The concept of the limit is a cornerstone of the mathematical branch of [[analysis]].

The concept of the '''limit''' is a cornerstone of the mathematical branch of [[analysis]] and captures the intuitive concept of "approaching a value".

 

==Limit of a Sequence==

==Limit of a Sequence==

Let <math>(a_n)_{n\in N} = (a_1, a_2, a_3, ...)</math> be a [[sequence]] of real numbers. We say that this sequence has a limit <math>a</math>, i.e.,

Let <math>(a_n)_{n\in N} = (a_1, a_2, a_3, ...)</math> be a [[sequence]] of [[real number]]s. We say that this sequence has a limit <math>a</math>, i.e.,

<math>

<math>

\lim_{n\to \infty} = a,  

\lim_{n\to \infty} = a,  

</math>

</math>

if for any <math>\epsilon > 0</math>, there exists a number  <math>N</math>, such that

if for any <math>\varepsilon > 0</math>, there exists a number  <math>N</math>, such that

<math>

<math>

|a_n - a| < \epsilon</math> for every <math>n > N</math>.  

|a_n - a| < \varepsilon</math> for every <math>n > N</math>. Intuitively, this means that if you have an infinitely small ball [[center]]ed at the limit point, then most of the points of the sequence are in that ball.

 

==Limit of a Function==

==Limit of a Function==

Let <math>f(x)</math> be a real valued [[function]] in one [[variable]]. We say that

Let <math>f(x)</math> be a real valued [[function]] in one [[variable]]. We say that

</math>

</math>

if for any error <math>\varepsilon</math>, we can find a sufficiently small neighborhood of <math>p</math> so that <math>f(x)</math> is within <math>\varepsilon</math> of the value <math>L</math> for any <math>x\neq p</math> in that neighborhood. One says that the limit of ''f(x)'' at ''p'' exists and is equal to ''L''.

if for any error <math>\varepsilon</math>, we can find a sufficiently small [[neighborhood]] of <math>p</math> so that <math>f(x)</math> is within <math>\varepsilon</math> of the value <math>L</math> for any <math>x\neq p</math> in that neighborhood. One says that the limit of ''f(x)'' at ''p'' exists and is equal to ''L''.

The standard, though more verbose, way of saying this is that for any <math>\varepsilon > 0</math> there exists a sufficiently small <math>\delta > 0</math> such that <math>|f(x)-L| <\varepsilon</math> whenever <math>0<|x-p|<\delta</math>.

The standard, though more verbose, way of saying this is that for any <math>\varepsilon > 0</math> there exists a sufficiently small <math>\delta > 0</math> such that <math>|f(x)-L| <\varepsilon</math> whenever <math>0<|x-p|<\delta</math>.

[[category:mathematics]]

[[category:mathematics]]