![X^2+sign(x)](https://upload.wikimedia.org/wikipedia/commons/b/be/X%5E2%2Bsign%28x%29.svg)
In
calculus
Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithm ...
, a one-sided limit refers to either one of the two
limits
Limit or Limits may refer to:
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* ''Limit'' (film), a South Korean film
* Limit (music), a way to characterize harmony
* "Limit" (song), a 2016 single by Luna Sea
* "Limits", a 2019 ...
of a
function
Function or functionality may refer to:
Computing
* Function key, a type of key on computer keyboards
* Function model, a structured representation of processes in a system
* Function object or functor or functionoid, a concept of object-oriente ...
of a
real
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...
variable
as
approaches a specified point either from the left or from the right.
The limit as
decreases in value approaching
(
approaches
"from the right" or "from above") can be denoted:
The limit as
increases in value approaching
(
approaches
"from the left" or "from below") can be denoted:
If the limit of
as
approaches
exists then the limits from the left and from the right both exist and are equal.
In some cases in which the limit
does not exist, the two one-sided limits nonetheless exist. Consequently, the limit as
approaches
is sometimes called a "two-sided limit".
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 definition
Definition
If
represents some
interval that is contained in the
domain
Domain may refer to:
Mathematics
*Domain of a function, the set of input values for which the (total) function is defined
**Domain of definition of a partial function
**Natural domain of a partial function
**Domain of holomorphy of a function
* Do ...
of
and if
is point in
then the right-sided limit as
approaches
can be rigorously defined as the value
that satisfies:
and the left-sided limit as
approaches
can be rigorously defined as the value
that satisfies:
We can represent the same thing more symbolically, as follows.
Let
represent an interval, where
, and
.
:
:
Intuition
In comparison to the formal definition for the
limit of a function
Although the function (sin ''x'')/''x'' is not defined at zero, as ''x'' becomes closer and closer to zero, (sin ''x'')/''x'' becomes arbitrarily close to 1. In other words, the limit of (sin ''x'')/''x'', as ''x'' approaches z ...
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:
:
To define a one-sided limit, we must modify this inequality. Note that the absolute distance between
and
is
.
For the limit from the right, we want
to be to the right of
, which means that
, so
is positive. From above,
is the distance between
and
. We want to bound this distance by our value of
, giving the inequality
. Putting together the inequalities
and
and using the
transitivity property of inequalities, we have the compound inequality
.
Similarly, for the limit from the left, we want
to be to the left of
, which means that
. In this case, it is
that is positive and represents the distance between
and
. Again, we want to bound this distance by our value of
, leading to the compound inequality
.
Now, when our value of
is in its desired interval, we expect that the value of
is also within its desired interval. The distance between
and
, the limiting value of the left sided limit, is
. Similarly, the distance between
and
, the limiting value of the right sided limit, is
. In both cases, we want to bound this distance by
, so we get the following:
for the left sided limit, and
for the right sided limit.
Examples
''Example 1'':
The limits from the left and from the right of
as
approaches
are
The reason why
is because
is always negative (since
means that
with all values of
satisfying
), which implies that
is always positive so that
diverges
[A limit that is equal to is said to verge to rather than verge to The same is true when a limit is equal to ] to
(and not to
) as
approaches
from the left.
Similarly,
since all values of
satisfy
(said differently,
is always positive) as
approaches
from the right, which implies that
is always negative so that
diverges to
![1 div (1 + 2 ** (-1 div x))](https://upload.wikimedia.org/wikipedia/commons/7/76/1_div_%281_%2B_2_%2A%2A_%28-1_div_x%29%29.svg)
''Example 2'':
One example of a function with different one-sided limits is
(cf. picture) where the limit from the left is
and the limit from the right is
To calculate these limits, first show that
(which is true because
)
so that consequently,
whereas
because the denominator diverges to infinity; that is, because
Since
the limit
does not exist.
Relation to topological definition of limit
The one-sided limit to a point
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
Alternatively, one may consider the domain with a
half-open interval topology
In mathematics, the lower limit topology or right half-open interval topology is a topology defined on the set \mathbb of real numbers; it is different from the standard topology on \mathbb (generated by the open intervals) and has a number of inte ...
.
Abel's theorem
A noteworthy theorem treating one-sided limits of certain
power series
In mathematics, a power series (in one variable) is an infinite series of the form
\sum_^\infty a_n \left(x - c\right)^n = a_0 + a_1 (x - c) + a_2 (x - c)^2 + \dots
where ''an'' represents the coefficient of the ''n''th term and ''c'' is a const ...
at the boundaries of their
intervals of convergence is
Abel's theorem
In mathematics, Abel's theorem for power series relates a limit of a power series to the sum of its coefficients. It is named after Norwegian mathematician Niels Henrik Abel.
Theorem
Let the Taylor series
G (x) = \sum_^\infty a_k x^k
be a powe ...
.
Notes
References
See also
*
Projectively extended real line
In real analysis, the projectively extended real line (also called the one-point compactification of the real line), is the extension of the set of the real numbers, \mathbb, by a point denoted . It is thus the set \mathbb\cup\ with the standar ...
*
Semi-differentiability In calculus, a branch of mathematics, the notions of one-sided differentiability and semi-differentiability of a real-valued function ''f'' of a real variable are weaker than differentiability. Specifically, the function ''f'' is said to be right di ...
*
Limit superior and limit inferior
In mathematics, the limit inferior and limit superior of a sequence can be thought of as limiting (that is, eventual and extreme) bounds on the sequence. They can be thought of in a similar fashion for a function (see limit of a function). For a ...
{{Calculus topics
Real analysis
Limits (mathematics)
Functions and mappings