In calculus, a one-sided limit refers to either one 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.[1][2]
The limit, as x decreases in value approaching a (x approaches a "from the right"[3] or "from above"), is denoted:[1][2]
\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(a+).
The limit, as x increases in value approaching a (x approaches a "from the left"[4][5] or "from below"), is denoted:[1][2]
\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(a-).
If the limits from the left and right both exist and are equal, then the limit of f(x) as x approaches a exists. Conversely, if the limit of f(x) as x approaches a exists, then the limits from left and right both exist and are equal. Consequently, the limit as x approaches a is sometimes called a "two-sided limit".[citation needed] It is denoted:
\lim_{x \to a} f(x).
In some cases in which the two-sided limit does not exist, the two individual one-sided limits nonetheless exist and they are then necessarily unequal.
It is possible for only one of the two one-sided limits to exist. It is also possible for neither of the two one-sided limits to exist.
Formal definition
Definition
If I represents some interval that is contained in the domain of a function f and if a is a point in I, then the right-sided limit as x approaches a can be rigorously defined as the value R that satisfies:[6]
for all
\varepsilon > 0there exists some\delta > 0such that for allx \in I, if0 < x - a < \deltathen|f(x) - R| < \varepsilon,
and the left-sided limit as x approaches a can be rigorously defined as the value L that satisfies:
for all
\varepsilon > 0there exists some\delta > 0such that for allx \in I, if0 < a - x < \deltathen|f(x) - L| < \varepsilon.
These definitions can be represented more symbolically as follows: Let I represent an interval, where I \subseteq \mathrm{domain}(f) and a \in I, then
\begin{align}
\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, \\
\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.
\end{align}
Intuition
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:
\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
.
To define a one-sided limit, we must modify this inequality. Note that the absolute distance between x and a is
|x - a| = |(-1)(-x + a)| = |(-1)(a - x)| = |(-1)||a - x| = |a - x|.
For the limit from the right, we want x to be to the right of a, which means that a < x, so x - a is positive. From above, x - a is the distance between x and a. We want to bound this distance by our value of \delta, giving the inequality x - a < \delta. Putting together the inequalities 0 < x - a and x - a < \delta and using the transitivity property of inequalities, we have the compound inequality 0 < x - a < \delta.
Similarly, for the limit from the left, we want x to be to the left of a, which means that x < a. In this case, it is a - x that is positive and represents the distance between x and a. Again, we want to bound this distance by our value of \delta, leading to the compound inequality 0 < a - x < \delta.
Now, when our value of x is in its desired interval, we expect that the value of f(x) is also within its desired interval. The distance between f(x) and L, the limiting value of the left sided limit, is |f(x) - L|. Similarly, the distance between f(x) and R, the limiting value of the right sided limit, is |f(x) - R|. In both cases, we want to bound this distance by \varepsilon, so we get the following: |f(x) - L| < \varepsilon for the left sided limit, and |f(x) - R| < \varepsilon for the right sided limit.
Examples
Example 1. The limits from the left and from the right of g(x) := - \frac{1}{x} as x approaches a := 0 are, respectively
\lim_{x \to 0^-} -\frac{1}{x} = + \infty \qquad \text{ and } \qquad \lim_{x \to 0^+} {-1/x} = - \infty.
The reason why \lim_{x \to 0^-} -\frac{1}{x} = + \infty is because x is always negative (since x \to 0^- means that x \to 0 with all values of x satisfying x < 0), which implies that - 1/x is always positive so that \lim_{x \to 0^-} -\frac{1}{x} diverges[note 1] to + \infty (and not to - \infty) as x approaches 0 from the left.
Similarly, \lim_{x \to 0^+} -\frac{1}{x} = - \infty since all values of x satisfy x > 0 (said differently, x is always positive) as x approaches 0 from the right, which implies that - 1/x is always negative so that \lim_{x \to 0^+} -\frac{1}{x} diverges to - \infty.
Example 2. One example of a function with different one-sided limits is f(x) = \frac{1}{1 + 2^{-1/x}}, where the limit from the left is \lim_{x \to 0^-} f(x) = 0 and the limit from the right is \lim_{x \to 0^+} f(x) = 1. To calculate these limits, first show that
\lim_{x \to 0^-} 2^{-1/x} = \infty \qquad \text{ and } \qquad \lim_{x \to 0^+} 2^{-1/x} = 0,
which is true because \lim_{x \to 0^-} {-1/x} = + \infty and \lim_{x \to 0^+} {-1/x} = - \infty
so that consequently,
\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
whereas \lim_{x \to 0^-} \frac{1}{1 + 2^{-1/x}} = 0 because the denominator diverges to infinity; that is, because \lim_{x \to 0^-} 1 + 2^{-1/x} = \infty. Since \lim_{x \to 0^-} f(x) \neq \lim_{x \to 0^+} f(x), the limit \lim_{x \to 0} f(x) does not exist.
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.[1][verification needed] Alternatively, one may consider the domain with a half-open interval topology.[citation needed]
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.[citation needed]
Notes
- ^ A limit that is equal to
\inftyis said to diverge to\inftyrather than converge to\infty.The same is true when a limit is equal to- \infty.
References
- ^ "One-sided limit - Encyclopedia of Mathematics". encyclopediaofmath.org. Archived from the original on 18 March 2023. Retrieved 7 August 2021.
- ^ Fridy, J. A. (24 January 2020). Introductory Analysis: The Theory of Calculus. Gulf Professional Publishing. p. 48. ISBN 978-0-12-267655-0. Retrieved 7 August 2021.
- ^ Hasan, Osman & Khayam, Syed (2014-01-02). "Towards Formal Linear Cryptanalysis using HOL4". Journal of Universal Computer Science. 20 (2): 209. doi:10.3217/jucs-020-02-0193. ISSN 0948-6968
- ^ Gasic, Andrei G. (2020-12-12). Phase Phenomena of Proteins in Living Matter (Thesis)
- ^ Brokate, Martin; Manchanda, Pammy; Siddiqi, Abul Hasan (2019), "Limit and Continuity", Calculus for Scientists and Engineers, Industrial and Applied Mathematics, Singapore: Springer Singapore, pp. 39–53, doi:10.1007/978-981-13-8464-6_2. ISBN 978-981-13-8463-9. S2CID 201484118, retrieved 2022-01-11
- ^ Swokowski, Earl W. (1979). Calculus with Analytic Geometry. 2nd ed. Taylor & Francis. p. 72–73. ISBN 978-0-87150-268-1.