Difference between revisions of "Limit (mathematics)"
m (wikify) |
(The "Limit of a Function" part is barely intelligible , IMO) |
||
| Line 1: | Line 1: | ||
| + | The concept of the limit is a cornerstone of the mathematical branch of [[analysis]]. | ||
| + | ==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., | ||
| + | <math> | ||
| + | \lim_{n\to \infty} = a, | ||
| + | </math> | ||
| + | if for any <math>\epsilon > 0</math>, there exists a number <math>N</math>, such that | ||
| + | <math> | ||
| + | |a_n - a| < \epsilon</math> for every <math>n > N</math>. | ||
| + | ==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 | ||
Revision as of 18:32, August 1, 2008
The concept of the limit is a cornerstone of the mathematical branch of analysis.
Limit of a Sequence
Let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (a_n)_{n\in N} = (a_1, a_2, a_3, ...)} be a sequence of real numbers. We say that this sequence has a limit , i.e., if for any , there exists a number , such that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |a_n - a| < \epsilon} for every Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n > N} .
Limit of a Function
Let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x)} be a real valued function in one variable. We say that
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lim_{x\to p} f(x) = L }
if for any error Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \varepsilon} , we can find a sufficiently small neighborhood of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p} so that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x)} is within Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \varepsilon} of the value Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L} for any Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x\neq p} 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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \varepsilon > 0} there exists a sufficiently small Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \delta > 0} such that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |f(x)-L| <\varepsilon} whenever Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0<|x-p|<\delta} .