One-sided limit
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In calculus, a one-sided limit refers to either one of the two limits of a function <math>f(x)</math> of a real variable <math>x</math> as <math>x</math> approaches a specified point either from the left or from the right.<ref name=":0">{{#invoke:citation/CS1|citation |CitationClass=web }}</ref><ref name=":1">Template:Cite book</ref>
The limit as <math>x</math> decreases in value approaching <math>a</math> (<math>x</math> approaches <math>a</math> "from the right"<ref>Template:Cite journal</ref> or "from above") can be denoted:<ref name=":0" /><ref name=":1" />
<math display=block>\lim_{x \to a^+}f(x) \quad \text{ or } \quad \lim_{x\,\downarrow\,a}\,f(x) \quad \text{ or } \quad \lim_{x \searrow a}\,f(x) \quad \text{ or } \quad f(x+)</math>
The limit as <math>x</math> increases in value approaching <math>a</math> (<math>x</math> approaches <math>a</math> "from the left"<ref>Template:Cite thesis</ref><ref>Template:Citation</ref> or "from below") can be denoted:<ref name=":0" /><ref name=":1" />
<math display=block>\lim_{x \to a^-}f(x) \quad \text{ or } \quad \lim_{x\,\uparrow\,a}\, f(x) \quad \text{ or } \quad \lim_{x \nearrow a}\,f(x) \quad \text{ or } \quad f(x-)</math>
If the limit of <math>f(x)</math> as <math>x</math> approaches <math>a</math> exists then the limits from the left and from the right both exist and are equal. In some cases in which the limit <math display=block>\lim_{x \to a} f(x)</math> does not exist, the two one-sided limits nonetheless exist. Consequently, the limit as <math>x</math> approaches <math>a</math> is sometimes called a "two-sided limit".Template:Citation needed
It is possible for exactly one of the two one-sided limits to exist (while the other does not exist). It is also possible for neither of the two one-sided limits to exist.
Formal definitionEdit
DefinitionEdit
If <math>I</math> represents some interval that is contained in the domain of <math>f</math> and if <math>a</math> is a point in <math>I</math> then the right-sided limit as <math>x</math> approaches <math>a</math> can be rigorously defined as the value <math>R</math> that satisfies:<ref>Template:Cite book</ref>Template:Verify source <math display=block>\text{for all } \varepsilon > 0\;\text{ there exists some } \delta > 0 \;\text{ such that for all } x \in I, \text{ if } \;0 < x - a < \delta \text{ then } |f(x) - R| < \varepsilon,</math> and the left-sided limit as <math>x</math> approaches <math>a</math> can be rigorously defined as the value <math>L</math> that satisfies: <math display=block>\text{for all } \varepsilon > 0\;\text{ there exists some } \delta > 0 \;\text{ such that for all } x \in I, \text{ if } \;0 < a - x < \delta \text{ then } |f(x) - L| < \varepsilon.</math>
We can represent the same thing more symbolically, as follows.
Let <math>I</math> represent an interval, where <math>I \subseteq \mathrm{domain}(f)</math>, and <math>a \in I </math>.
- <math display=block>
\lim_{x \to a^{+}} f(x) = R
~~~ \iff ~~~
(\forall \varepsilon \in \mathbb{R}_{+}, \exists \delta \in \mathbb{R}_{+}, \forall x \in I,
(0 < x - a < \delta \longrightarrow | f(x) - R | < \varepsilon))
</math>
- <math display=block>
\lim_{x \to a^{-}} f(x) = L
~~~ \iff ~~~
(\forall \varepsilon \in \mathbb{R}_{+}, \exists \delta \in \mathbb{R}_{+}, \forall x \in I,
(0 < a - x < \delta \longrightarrow | f(x) - L | < \varepsilon))
</math>
IntuitionEdit
In comparison to the formal definition for the limit of a function at a point, the one-sided limit (as the name would suggest) only deals with input values to one side of the approached input value.
For reference, the formal definition for the limit of a function at a point is as follows:
- <math display=block>
\lim_{x \to a} f(x) = L
~~~ \iff ~~~
\forall \varepsilon \in \mathbb{R}_{+}, \exists \delta \in \mathbb{R}_{+}, \forall x \in I,
0 < |x - a| < \delta \implies | f(x) - L | < \varepsilon
.</math>
To define a one-sided limit, we must modify this inequality. Note that the absolute distance between <math>x</math> and <math>a</math> is
<math display=block>|x - a| = |(-1)(-x + a)| = |(-1)(a - x)| = |(-1)||a - x| = |a - x|.</math>
For the limit from the right, we want <math>x</math> to be to the right of <math>a</math>, which means that <math>a < x</math>, so <math>x - a</math> is positive. From above, <math>x - a</math> is the distance between <math>x</math> and <math>a</math>. We want to bound this distance by our value of <math>\delta</math>, giving the inequality <math>x - a < \delta</math>. Putting together the inequalities <math>0 < x - a</math> and <math>x - a < \delta</math> and using the transitivity property of inequalities, we have the compound inequality <math>0 < x - a < \delta </math>.
Similarly, for the limit from the left, we want <math>x</math> to be to the left of <math>a</math>, which means that <math>x < a</math>. In this case, it is <math>a - x</math> that is positive and represents the distance between <math>x</math> and <math>a</math>. Again, we want to bound this distance by our value of <math>\delta</math>, leading to the compound inequality <math>0 < a - x < \delta </math>.
Now, when our value of <math>x</math> is in its desired interval, we expect that the value of <math>f(x)</math> is also within its desired interval. The distance between <math>f(x)</math> and <math>L</math>, the limiting value of the left sided limit, is <math>|f(x) - L|</math>. Similarly, the distance between <math>f(x)</math> and <math>R</math>, the limiting value of the right sided limit, is <math>|f(x) - R|</math>. In both cases, we want to bound this distance by <math>\varepsilon</math>, so we get the following: <math>|f(x) - L| < \varepsilon</math> for the left sided limit, and <math>|f(x) - R| < \varepsilon</math> for the right sided limit.
ExamplesEdit
Example 1: The limits from the left and from the right of <math>g(x) := - \frac{1}{x}</math> as <math>x</math> approaches <math>a := 0</math> are <math display=block>\lim_{x \to 0^-} {-1/x} = + \infty \qquad \text{ and } \qquad \lim_{x \to 0^+} {-1/x} = - \infty</math> The reason why <math>\lim_{x \to 0^-} {-1/x} = + \infty</math> is because <math>x</math> is always negative (since <math>x \to 0^-</math> means that <math>x \to 0</math> with all values of <math>x</math> satisfying <math>x < 0</math>), which implies that <math>- 1/x</math> is always positive so that <math>\lim_{x \to 0^-} {-1/x}</math> diverges<ref group=note>A limit that is equal to <math>\infty</math> is said to Template:Emverge to <math>\infty</math> rather than Template:Emverge to <math>\infty.</math> The same is true when a limit is equal to <math>- \infty.</math></ref> to <math>+ \infty</math> (and not to <math>- \infty</math>) as <math>x</math> approaches <math>0</math> from the left. Similarly, <math>\lim_{x \to 0^+} {-1/x} = - \infty</math> since all values of <math>x</math> satisfy <math>x > 0</math> (said differently, <math>x</math> is always positive) as <math>x</math> approaches <math>0</math> from the right, which implies that <math>- 1/x</math> is always negative so that <math>\lim_{x \to 0^+} {-1/x}</math> diverges to <math>- \infty.</math>
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Example 2: One example of a function with different one-sided limits is <math>f(x) = \frac{1}{1 + 2^{-1/x}},</math> (cf. picture) where the limit from the left is <math>\lim_{x \to 0^-} f(x) = 0</math> and the limit from the right is <math>\lim_{x \to 0^+} f(x) = 1.</math> To calculate these limits, first show that <math display=block>\lim_{x \to 0^-} 2^{-1/x} = \infty \qquad \text{ and } \qquad \lim_{x \to 0^+} 2^{-1/x} = 0</math> (which is true because <math>\lim_{x \to 0^-} {-1/x} = + \infty \text{ and } \lim_{x \to 0^+} {-1/x} = - \infty</math>) so that consequently, <math display=block>\lim_{x \to 0^+} \frac{1}{1 + 2^{-1/x}} = \frac{1}{1 + \displaystyle\lim_{x \to 0^+} 2^{-1/x}} = \frac{1}{1 + 0} = 1</math> whereas <math>\lim_{x \to 0^-} \frac{1}{1 + 2^{-1/x}} = 0</math> because the denominator diverges to infinity; that is, because <math>\lim_{x \to 0^-} 1 + 2^{-1/x} = \infty.</math> Since <math>\lim_{x \to 0^-} f(x) \neq \lim_{x \to 0^+} f(x),</math> the limit <math>\lim_{x \to 0} f(x)</math> does not exist.
Relation to topological definition of limitEdit
The one-sided limit to a point <math>p</math> 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 <math>p.</math><ref name=":0" />Template:Verify source Alternatively, one may consider the domain with a half-open interval topology.Template:Citation needed
Abel's theoremEdit
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A noteworthy theorem treating one-sided limits of certain power series at the boundaries of their intervals of convergence is Abel's theorem.Template:Citation needed