- Source: One-sided limit
In calculus, a one-sided limit refers to either one of the two limits of a function
f
(
x
)
{\displaystyle f(x)}
of a real variable
x
{\displaystyle x}
as
x
{\displaystyle x}
approaches a specified point either from the left or from the right.
The limit as
x
{\displaystyle x}
decreases in value approaching
a
{\displaystyle a}
(
x
{\displaystyle x}
approaches
a
{\displaystyle a}
"from the right" or "from above") can be denoted:
lim
x
→
a
+
f
(
x
)
or
lim
x
↓
a
f
(
x
)
or
lim
x
↘
a
f
(
x
)
or
f
(
x
+
)
{\displaystyle \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+)}
The limit as
x
{\displaystyle x}
increases in value approaching
a
{\displaystyle a}
(
x
{\displaystyle x}
approaches
a
{\displaystyle a}
"from the left" or "from below") can be denoted:
lim
x
→
a
−
f
(
x
)
or
lim
x
↑
a
f
(
x
)
or
lim
x
↗
a
f
(
x
)
or
f
(
x
−
)
{\displaystyle \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-)}
If the limit of
f
(
x
)
{\displaystyle f(x)}
as
x
{\displaystyle x}
approaches
a
{\displaystyle a}
exists then the limits from the left and from the right both exist and are equal. In some cases in which the limit
lim
x
→
a
f
(
x
)
{\displaystyle \lim _{x\to a}f(x)}
does not exist, the two one-sided limits nonetheless exist. Consequently, the limit as
x
{\displaystyle x}
approaches
a
{\displaystyle a}
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
I
{\displaystyle I}
represents some interval that is contained in the domain of
f
{\displaystyle f}
and if
a
{\displaystyle a}
is a point in
I
{\displaystyle I}
then the right-sided limit as
x
{\displaystyle x}
approaches
a
{\displaystyle a}
can be rigorously defined as the value
R
{\displaystyle R}
that satisfies:
for all
ε
>
0
there exists some
δ
>
0
such that for all
x
∈
I
,
if
0
<
x
−
a
<
δ
then
|
f
(
x
)
−
R
|
<
ε
,
{\displaystyle {\text{for all }}\varepsilon >0\;{\text{ there exists some }}\delta >0\;{\text{ such that for all }}x\in I,{\text{ if }}\;0
and the left-sided limit as
x
{\displaystyle x}
approaches
a
{\displaystyle a}
can be rigorously defined as the value
L
{\displaystyle L}
that satisfies:
for all
ε
>
0
there exists some
δ
>
0
such that for all
x
∈
I
,
if
0
<
a
−
x
<
δ
then
|
f
(
x
)
−
L
|
<
ε
.
{\displaystyle {\text{for all }}\varepsilon >0\;{\text{ there exists some }}\delta >0\;{\text{ such that for all }}x\in I,{\text{ if }}\;0
We can represent the same thing more symbolically, as follows.
Let
I
{\displaystyle I}
represent an interval, where
I
⊆
d
o
m
a
i
n
(
f
)
{\displaystyle I\subseteq \mathrm {domain} (f)}
, and
a
∈
I
{\displaystyle a\in I}
.
lim
x
→
a
+
f
(
x
)
=
R
⟺
(
∀
ε
∈
R
+
,
∃
δ
∈
R
+
,
∀
x
∈
I
,
(
0
<
x
−
a
<
δ
⟶
|
f
(
x
)
−
R
|
<
ε
)
)
{\displaystyle \lim _{x\to a^{+}}f(x)=R~~~\iff ~~~(\forall \varepsilon \in \mathbb {R} _{+},\exists \delta \in \mathbb {R} _{+},\forall x\in I,(0
lim
x
→
a
−
f
(
x
)
=
L
⟺
(
∀
ε
∈
R
+
,
∃
δ
∈
R
+
,
∀
x
∈
I
,
(
0
<
a
−
x
<
δ
⟶
|
f
(
x
)
−
L
|
<
ε
)
)
{\displaystyle \lim _{x\to a^{-}}f(x)=L~~~\iff ~~~(\forall \varepsilon \in \mathbb {R} _{+},\exists \delta \in \mathbb {R} _{+},\forall x\in I,(0
= 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
→
a
f
(
x
)
=
L
⟺
∀
ε
∈
R
+
,
∃
δ
∈
R
+
,
∀
x
∈
I
,
0
<
|
x
−
a
|
<
δ
⟹
|
f
(
x
)
−
L
|
<
ε
.
{\displaystyle \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
{\displaystyle x}
and
a
{\displaystyle a}
is
|
x
−
a
|
=
|
(
−
1
)
(
−
x
+
a
)
|
=
|
(
−
1
)
(
a
−
x
)
|
=
|
(
−
1
)
|
|
a
−
x
|
=
|
a
−
x
|
.
{\displaystyle |x-a|=|(-1)(-x+a)|=|(-1)(a-x)|=|(-1)||a-x|=|a-x|.}
For the limit from the right, we want
x
{\displaystyle x}
to be to the right of
a
{\displaystyle a}
, which means that
a
<
x
{\displaystyle a
, so
x
−
a
{\displaystyle x-a}
is positive. From above,
x
−
a
{\displaystyle x-a}
is the distance between
x
{\displaystyle x}
and
a
{\displaystyle a}
. We want to bound this distance by our value of
δ
{\displaystyle \delta }
, giving the inequality
x
−
a
<
δ
{\displaystyle x-a<\delta }
. Putting together the inequalities
0
<
x
−
a
{\displaystyle 0
and
x
−
a
<
δ
{\displaystyle x-a<\delta }
and using the transitivity property of inequalities, we have the compound inequality
0
<
x
−
a
<
δ
{\displaystyle 0
.
Similarly, for the limit from the left, we want
x
{\displaystyle x}
to be to the left of
a
{\displaystyle a}
, which means that
x
<
a
{\displaystyle x
. In this case, it is
a
−
x
{\displaystyle a-x}
that is positive and represents the distance between
x
{\displaystyle x}
and
a
{\displaystyle a}
. Again, we want to bound this distance by our value of
δ
{\displaystyle \delta }
, leading to the compound inequality
0
<
a
−
x
<
δ
{\displaystyle 0
.
Now, when our value of
x
{\displaystyle x}
is in its desired interval, we expect that the value of
f
(
x
)
{\displaystyle f(x)}
is also within its desired interval. The distance between
f
(
x
)
{\displaystyle f(x)}
and
L
{\displaystyle L}
, the limiting value of the left sided limit, is
|
f
(
x
)
−
L
|
{\displaystyle |f(x)-L|}
. Similarly, the distance between
f
(
x
)
{\displaystyle f(x)}
and
R
{\displaystyle R}
, the limiting value of the right sided limit, is
|
f
(
x
)
−
R
|
{\displaystyle |f(x)-R|}
. In both cases, we want to bound this distance by
ε
{\displaystyle \varepsilon }
, so we get the following:
|
f
(
x
)
−
L
|
<
ε
{\displaystyle |f(x)-L|<\varepsilon }
for the left sided limit, and
|
f
(
x
)
−
R
|
<
ε
{\displaystyle |f(x)-R|<\varepsilon }
for the right sided limit.
Examples
Example 1:
The limits from the left and from the right of
g
(
x
)
:=
−
1
x
{\displaystyle g(x):=-{\frac {1}{x}}}
as
x
{\displaystyle x}
approaches
a
:=
0
{\displaystyle a:=0}
are
lim
x
→
0
−
−
1
/
x
=
+
∞
and
lim
x
→
0
+
−
1
/
x
=
−
∞
{\displaystyle \lim _{x\to 0^{-}}{-1/x}=+\infty \qquad {\text{ and }}\qquad \lim _{x\to 0^{+}}{-1/x}=-\infty }
The reason why
lim
x
→
0
−
−
1
/
x
=
+
∞
{\displaystyle \lim _{x\to 0^{-}}{-1/x}=+\infty }
is because
x
{\displaystyle x}
is always negative (since
x
→
0
−
{\displaystyle x\to 0^{-}}
means that
x
→
0
{\displaystyle x\to 0}
with all values of
x
{\displaystyle x}
satisfying
x
<
0
{\displaystyle x<0}
), which implies that
−
1
/
x
{\displaystyle -1/x}
is always positive so that
lim
x
→
0
−
−
1
/
x
{\displaystyle \lim _{x\to 0^{-}}{-1/x}}
diverges to
+
∞
{\displaystyle +\infty }
(and not to
−
∞
{\displaystyle -\infty }
) as
x
{\displaystyle x}
approaches
0
{\displaystyle 0}
from the left.
Similarly,
lim
x
→
0
+
−
1
/
x
=
−
∞
{\displaystyle \lim _{x\to 0^{+}}{-1/x}=-\infty }
since all values of
x
{\displaystyle x}
satisfy
x
>
0
{\displaystyle x>0}
(said differently,
x
{\displaystyle x}
is always positive) as
x
{\displaystyle x}
approaches
0
{\displaystyle 0}
from the right, which implies that
−
1
/
x
{\displaystyle -1/x}
is always negative so that
lim
x
→
0
+
−
1
/
x
{\displaystyle \lim _{x\to 0^{+}}{-1/x}}
diverges to
−
∞
.
{\displaystyle -\infty .}
Example 2:
One example of a function with different one-sided limits is
f
(
x
)
=
1
1
+
2
−
1
/
x
,
{\displaystyle f(x)={\frac {1}{1+2^{-1/x}}},}
(cf. picture) where the limit from the left is
lim
x
→
0
−
f
(
x
)
=
0
{\displaystyle \lim _{x\to 0^{-}}f(x)=0}
and the limit from the right is
lim
x
→
0
+
f
(
x
)
=
1.
{\displaystyle \lim _{x\to 0^{+}}f(x)=1.}
To calculate these limits, first show that
lim
x
→
0
−
2
−
1
/
x
=
∞
and
lim
x
→
0
+
2
−
1
/
x
=
0
{\displaystyle \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
→
0
−
−
1
/
x
=
+
∞
and
lim
x
→
0
+
−
1
/
x
=
−
∞
{\displaystyle \lim _{x\to 0^{-}}{-1/x}=+\infty {\text{ and }}\lim _{x\to 0^{+}}{-1/x}=-\infty }
)
so that consequently,
lim
x
→
0
+
1
1
+
2
−
1
/
x
=
1
1
+
lim
x
→
0
+
2
−
1
/
x
=
1
1
+
0
=
1
{\displaystyle \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
→
0
−
1
1
+
2
−
1
/
x
=
0
{\displaystyle \lim _{x\to 0^{-}}{\frac {1}{1+2^{-1/x}}}=0}
because the denominator diverges to infinity; that is, because
lim
x
→
0
−
1
+
2
−
1
/
x
=
∞
.
{\displaystyle \lim _{x\to 0^{-}}1+2^{-1/x}=\infty .}
Since
lim
x
→
0
−
f
(
x
)
≠
lim
x
→
0
+
f
(
x
)
,
{\displaystyle \lim _{x\to 0^{-}}f(x)\neq \lim _{x\to 0^{+}}f(x),}
the limit
lim
x
→
0
f
(
x
)
{\displaystyle \lim _{x\to 0}f(x)}
does not exist.
Relation to topological definition of limit
The one-sided limit to a point
p
{\displaystyle 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
.
{\displaystyle 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.
Notes
References
See also
Projectively extended real line
Semi-differentiability
Limit superior and limit inferior
Kata Kunci Pencarian:
- One-sided limit
- Limit (mathematics)
- Limit of a function
- One-sided
- Limit
- Semi-differentiability
- Limit inferior and limit superior
- Classification of discontinuities
- List of calculus topics
- P-value
