Difference between revisions of "Real Function Limits:One-Sided"
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| − | In | + | In calculus, a '''one-sided limit''' is either of the two limits of a function ''f''(''x'') of a real variable ''x'' as ''x'' approaches a specified point either from the left or from the right. |
| − | The limit as ''x'' decreases in value approaching ''a'' (''x'' approaches ''a'' | + | The limit as ''x'' decreases in value approaching ''a'' (''x'' approaches ''a'' ''from the right'' or "from above") can be denoted: |
| − | :<math>\lim_{x \to a^+}f(x)\ </math> or <math> \lim_{x\,\downarrow\,a}\,f(x)</math> or <math> \lim_{x \searrow a}\,f(x)</math> or <math>\lim_{x \underset{>}{\to} a}f(x)</math> | + | :<math>\lim_{x \to a^+}f(x)\ </math> or <math> \lim_{x\,\downarrow\,a}\,f(x)</math> or <math> \lim_{x \searrow a}\,f(x)</math> or <math>\lim_{x \underset{>}{\to} a}f(x)</math> |
| − | The limit as ''x'' increases in value approaching ''a'' (''x'' approaches ''a'' | + | The limit as ''x'' increases in value approaching ''a'' (''x'' approaches ''a'' "from the left" or "from below") can be denoted: |
| − | :<math>\lim_{x \to a^-}f(x)\ </math> or <math> \lim_{x\,\uparrow\,a}\, f(x)</math> or <math> \lim_{x \nearrow a}\,f(x)</math> or <math>\lim_{x \underset{<}{\to} a}f(x)</math> | + | :<math>\lim_{x \to a^-}f(x)\ </math> or <math> \lim_{x\,\uparrow\,a}\, f(x)</math> or <math> \lim_{x \nearrow a}\,f(x)</math> or <math>\lim_{x \underset{<}{\to} a}f(x)</math> |
| − | + | In probability theory it is common to use the short notation: | |
| − | :<math>f(x-)</math> for the left limit and <math>f(x+)</math> for the right limit. | + | :<math>f(x-)</math> for the left limit and <math>f(x+)</math> for the right limit. |
| − | The two one-sided limits exist and are equal if the limit of ''f''(''x'') as ''x'' approaches ''a'' exists. | + | The two one-sided limits exist and are equal if the limit of ''f''(''x'') as ''x'' approaches ''a'' exists. In some cases in which the limit |
:<math>\lim_{x\to a} f(x)\,</math> | :<math>\lim_{x\to a} f(x)\,</math> | ||
| − | does not exist, the two one-sided limits nonetheless exist. Consequently, the limit as ''x'' approaches ''a'' is sometimes called a "two-sided limit". | + | does not exist, the two one-sided limits nonetheless exist. Consequently, the limit as ''x'' approaches ''a'' is sometimes called a "two-sided limit". |
| − | In some cases one of the two one-sided limits exists and the other does not, and in some cases neither exists. | + | In some cases one of the two one-sided limits exists and the other does not, and in some cases neither exists. |
The right-sided limit can be rigorously defined as | The right-sided limit can be rigorously defined as | ||
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:<math>\forall\varepsilon > 0\;\exists \delta >0 \;\forall x \in I \;(0 < a - x < \delta \Rightarrow |f(x) - L|<\varepsilon),</math> | :<math>\forall\varepsilon > 0\;\exists \delta >0 \;\forall x \in I \;(0 < a - x < \delta \Rightarrow |f(x) - L|<\varepsilon),</math> | ||
| − | where {{mvar|I}} represents some | + | where {{mvar|I}} represents some interval that is within the domain of {{mvar|f}}. |
==Examples== | ==Examples== | ||
| Line 40: | Line 40: | ||
whereas | whereas | ||
| − | :<math>\lim_{x \to 0^-}{1 \over 1 + 2^{-1/x}} = 0.</math> | + | :<math>\lim_{x \to 0^-}{1 \over 1 + 2^{-1/x}} = 0.</math> |
==Relation to topological definition of limit== | ==Relation to topological definition of limit== | ||
| − | The one-sided limit to a point ''p'' corresponds to the | + | The one-sided limit to a point ''p'' corresponds to the general definition of limit, with the domain of the function restricted to one side, by either allowing that the function domain is a subset of the topological space, or by considering a one-sided subspace, including ''p''. Alternatively, one may consider the domain with a half-open interval topology. |
==Abel's theorem== | ==Abel's theorem== | ||
| − | A noteworthy theorem treating one-sided limits of certain | + | A noteworthy theorem treating one-sided limits of certain power series at the boundaries of their intervals of convergence is Abel's theorem. |
| − | == | + | == Licensing == |
| − | * [https://en.wikipedia.org/wiki/One-sided_limit One-sided limit | + | Content obtained and/or adapted from: |
| + | * [https://en.wikipedia.org/wiki/One-sided_limit One-sided limit, Wikipedia] under a CC BY-SA license | ||
Latest revision as of 16:13, 6 November 2021
In calculus, a one-sided limit is either of the two limits of a function f(x) of a real variable x as x approaches a specified point either from the left or from the right.
The limit as x decreases in value approaching a (x approaches a from the right or "from above") can be denoted:
- or or or
The limit as x increases in value approaching a (x approaches a "from the left" or "from below") can be denoted:
- or or or
In probability theory it is common to use the short notation:
- for the left limit and for the right limit.
The two one-sided limits exist and are equal if the limit of f(x) as x approaches a exists. In some cases in which the limit
does not exist, the two one-sided limits nonetheless exist. Consequently, the limit as x approaches a is sometimes called a "two-sided limit".
In some cases one of the two one-sided limits exists and the other does not, and in some cases neither exists.
The right-sided limit can be rigorously defined as
and the left-sided limit can be rigorously defined as
where I represents some interval that is within the domain of f.
Examples
).svg)
One example of a function with different one-sided limits is the following (cf. picture):
whereas
Relation to topological definition of limit
The one-sided limit to a point p corresponds to the general definition of limit, with the domain of the function restricted to one side, by either allowing that the function domain is a subset of the topological space, or by considering a one-sided subspace, including p. Alternatively, one may consider the domain with a half-open interval topology.
Abel's theorem
A noteworthy theorem treating one-sided limits of certain power series at the boundaries of their intervals of convergence is Abel's theorem.
Licensing
Content obtained and/or adapted from:
- One-sided limit, Wikipedia under a CC BY-SA license
