Limit value (function)

In the mathematics of the designated limit or limit of a function at a certain point that value, which approximates the function in the vicinity of the site in question. However, such a limit does not exist in all cases. If the limit value exists , the function converges , otherwise it diverges . The concept of limit values ​​was formalized in the 19th century. It is one of the most important concepts in calculus .

Formal definition of the limit of a real function

The limit of the function f for x against p is equal to L if and only if for every ε> 0 there is a δ> 0, so that for all x with 0 <| x - p | <δ also | f ( x ) - L | <ε holds.

The symbol , read “ Limes f from x for x against p ”, denotes the limit of the real function for the limit transition of the variables against . It can be either a real number or one of the symbolic values and . In the first case it does not necessarily have to be in the domain of , but it has to be an accumulation point of , i. That is, in every neighborhood of there must be an infinite number of elements of . In the case of or , the definition range must be unlimited from above or below. ${\displaystyle \lim _{x\to p}f(x)}$ ${\displaystyle f}$ ${\displaystyle x}$ ${\displaystyle p}$ ${\displaystyle p}$ ${\displaystyle +\infty }$ ${\displaystyle -\infty }$ ${\displaystyle p}$ ${\displaystyle D}$ ${\displaystyle f}$ ${\displaystyle D}$ ${\displaystyle p}$ ${\displaystyle D}$ ${\displaystyle p=\infty }$ ${\displaystyle p=-\infty }$ ${\displaystyle f}$

See also : Asymptotic behavior

Accordingly, there are several different definitions of the term Limes:

Argument finite, limit finite

• Definition: Let be a subset of and an accumulation point of . The function has for the limit , if there is one ( generally dependent) for each (no matter how small) , so that for all values ​​from the domain of that meet the condition also applies. ${\displaystyle X}$ ${\displaystyle \mathbb {R} }$ ${\displaystyle p\in \mathbb {R} }$ ${\displaystyle X}$ ${\displaystyle f\colon X\to \mathbb {R} }$ ${\displaystyle x\to p}$ ${\displaystyle L}$ ${\displaystyle \varepsilon >0}$ ${\displaystyle \varepsilon }$ ${\displaystyle \delta >0}$ ${\displaystyle x}$ ${\displaystyle X}$ ${\displaystyle f}$ ${\displaystyle 0<|x-p|<\delta }$ ${\displaystyle |f(x)-L|<\varepsilon }$

Expressed qualitatively, the definition means: The difference between the function value and the limit becomes arbitrarily small if you choose close enough to . ${\displaystyle f(x)}$ ${\displaystyle L}$ ${\displaystyle x}$ ${\displaystyle p}$

It should be noted that it does not matter what value the function has at that point ; the function does not even need to be defined at the point . The only decisive factor is the behavior of in the dotted surroundings of . However, some authors use a definition with surroundings that are not dotted; see the section “ Newer limit value concept ”. ${\displaystyle f}$ ${\displaystyle p}$ ${\displaystyle p}$ ${\displaystyle f}$ ${\displaystyle p}$

In contrast to the formulation used by Augustin-Louis Cauchy that “the function approaches the limit value”, it is not a variable that “runs”, but simply an element of a given set. This static ε-δ definition used today essentially goes back to Karl Weierstrass and placed the concept of limit values ​​on a solid mathematical foundation, the so-called epsilontics . ${\displaystyle x}$

Example: ${\displaystyle \lim _{x\to 1}{\frac {x^{2}-1}{x-1}}=\lim _{x\to 1}{\frac {(x-1)(x+1)}{x-1}}=\lim _{x\to 1}x+1=2}$

Argument finite, limit infinite

• Definition: The function has the limit for (with ) the limit if there is a (generally dependent) for every real number ( no matter how large) , so that for any values ​​from the definition range of that satisfy the condition is also fulfilled . ${\displaystyle f}$ ${\displaystyle x\to p}$ ${\displaystyle p\in \mathbb {R} }$ ${\displaystyle +\infty }$ ${\displaystyle T}$ ${\displaystyle T}$ ${\displaystyle \delta >0}$ ${\displaystyle x}$ ${\displaystyle f}$ ${\displaystyle 0<|x-p|<\delta }$ ${\displaystyle f(x)>T}$

In this case one calls for against certain divergent . ${\displaystyle \lim _{x\to p}f(x)=\infty }$ ${\displaystyle f}$ ${\displaystyle x}$ ${\displaystyle p}$

The case of the limit value is defined accordingly . ${\displaystyle -\infty }$

Example: ${\displaystyle \lim _{x\to 0}{\frac {1}{x^{2}}}=\infty }$

Argument infinite, limit finite

• Definition: The function has for the limit if there is a real number (generally dependent on) for each (no matter how small) , so that for any -values ​​from the definition range of that satisfy the condition is also fulfilled. ${\displaystyle f}$ ${\displaystyle x\to +\infty }$ ${\displaystyle L}$ ${\displaystyle \varepsilon >0}$ ${\displaystyle \varepsilon }$ ${\displaystyle S}$ ${\displaystyle x}$ ${\displaystyle f}$ ${\displaystyle x>S}$ ${\displaystyle |f(x)-L|<\varepsilon }$

In this case one calls for convergent towards infinity . ${\displaystyle \lim _{x\to \infty }f(x)=L}$ ${\displaystyle f}$ ${\displaystyle x}$

Limit values ​​of type or can be defined accordingly . ${\displaystyle x\to -\infty }$ ${\displaystyle L\in \{-\infty ,+\infty \}}$

Example: ${\displaystyle \lim _{x\to \infty }{\frac {x}{x+1}}=1}$

Definition with the help of sequences

In the real numbers, an accumulation point can be characterized as follows:

Be a subset of and . is an accumulation point if and only if there is a sequence with which fulfills, see limit value (sequence) . ${\displaystyle D}$ ${\displaystyle \mathbb {R} }$ ${\displaystyle p\in \mathbb {R} }$ ${\displaystyle p}$ ${\displaystyle D}$ ${\displaystyle (x_{n})_{n\in \mathbb {N} }}$ ${\displaystyle x_{n}\in D\setminus \{p\}}$ ${\displaystyle \lim _{n\to \infty }x_{n}=p}$

An alternative limit value definition can be formulated with this property:

• Definition: Let be a function, an accumulation point of and . Then we define: if and only if for every sequence with and where: . ${\displaystyle f\colon D\to \mathbb {R} }$ ${\displaystyle p}$ ${\displaystyle D}$ ${\displaystyle L\in \mathbb {R} \cup \{\pm \infty \}}$ ${\displaystyle \lim _{x\to p}f(x)=L}$ ${\displaystyle (x_{n})_{n\in \mathbb {N} }}$ ${\displaystyle x_{n}\in D\setminus \{p\}}$ ${\displaystyle \lim _{n\to \infty }x_{n}=p}$ ${\displaystyle \lim _{n\to \infty }f(x_{n})=L}$

As soon as you allow the accumulation point as a limit value in the definition, you can also define and . ${\displaystyle \pm \infty }$ ${\displaystyle \lim _{x\to \infty }f(x)}$ ${\displaystyle \lim _{x\to -\infty }f(x)}$

It can be shown that the - definition of the limit value is equivalent to the following definition. ${\displaystyle \varepsilon }$ ${\displaystyle \delta }$

Unilateral limits

definition

Let be a subset of and an accumulation point of . The function has for the limit , if there is one ( generally dependent) for each (no matter how small) , so that for all values ​​from the domain of that satisfy the condition also applies. ${\displaystyle X}$ ${\displaystyle \mathbb {R} }$ ${\displaystyle p\in \mathbb {R} }$ ${\displaystyle X\cap (p,\infty )}$ ${\displaystyle f\colon X\to \mathbb {R} }$ ${\displaystyle x\to p+}$ ${\displaystyle L}$ ${\displaystyle \varepsilon >0}$ ${\displaystyle \varepsilon }$ ${\displaystyle \delta >0}$ ${\displaystyle x}$ ${\displaystyle X}$ ${\displaystyle f}$ ${\displaystyle 0<x-p<\delta }$ ${\displaystyle |f(x)-L|<\varepsilon }$

In this case we call for from the right to convergent . ${\displaystyle \lim _{x\to p+}f(x)=L}$ ${\displaystyle f}$ ${\displaystyle x}$ ${\displaystyle p}$

Limit values ​​of the type or for are defined accordingly . ${\displaystyle x\to p-}$ ${\displaystyle L\in \{-\infty ,+\infty \}}$

Examples

function	right-hand limit	left-hand limit	limit value on both sides
${\displaystyle \operatorname {sgn}(x)}$	${\displaystyle \lim _{x\to 0+}\operatorname {sgn}(x)=+1}$	${\displaystyle \lim _{x\to 0-}\operatorname {sgn}(x)=-1}$	does not exist
${\displaystyle {\frac {1}{x}}}$	${\displaystyle \lim _{x\to 0+}{\frac {1}{x}}=+\infty }$	${\displaystyle \lim _{x\to 0-}{\frac {1}{x}}=-\infty }$	does not exist
${\displaystyle {\frac {1}{|x|}}}$	${\displaystyle \lim _{x\to 0+}{\frac {1}{|x|}}=+\infty }$	${\displaystyle \lim _{x\to 0-}{\frac {1}{|x|}}=+\infty }$	${\displaystyle \lim _{x\to 0}{\frac {1}{|x|}}=+\infty }$

notation

right-hand limit	${\displaystyle \lim _{x\to p+}f(x)}$	${\displaystyle \lim _{x\to p+0}f(x)}$	${\displaystyle \lim _{x\downarrow p}f(x)}$	${\displaystyle \lim _{x\searrow p}f(x)}$	${\displaystyle \lim _{x\to p \atop x>p}f(x)}$	${\displaystyle f(p+)}$
left-hand limit	${\displaystyle \lim _{x\to p-}f(x)}$	${\displaystyle \lim _{x\to p-0}f(x)}$	${\displaystyle \lim _{x\uparrow p}f(x)}$	${\displaystyle \lim _{x\nearrow p}f(x)}$	${\displaystyle \lim _{x\to p \atop x<p}f(x)}$	${\displaystyle f(p-)}$

Unilateral and bilateral limit value

In order to avoid confusion, one speaks in the case of sometimes of the bilateral limit value. If there is an accumulation point of and of , then: ${\displaystyle \lim _{x\to p}f(x)}$ ${\displaystyle p}$ ${\displaystyle X\cap (p,\infty )}$ ${\displaystyle X\cap (-\infty ,p)}$

${\displaystyle \lim _{x\rightarrow p}f(x)}$exists if and only if the two one-sided limit values and exist and match. In this case the equality applies . ${\displaystyle \lim _{x\nearrow p}f(x)}$ ${\displaystyle \lim _{x\searrow p}f(x)}$ ${\displaystyle \lim _{x\rightarrow p}f(x)=\lim _{x\nearrow p}f(x)=\lim _{x\searrow p}f(x)}$

And exactly when is defined in the point and applies, is continuous at the point . ${\displaystyle f}$ ${\displaystyle p}$ ${\displaystyle \lim _{x\rightarrow p}f(x)=f(p)}$ ${\displaystyle f}$ ${\displaystyle p}$

Limit sets

Let , and two real-valued functions, whose limit values and exist, where and is an accumulation point of the extended real numbers . Then the following limit values ​​also exist and can be calculated as indicated: ${\displaystyle D\subseteq \mathbb {R} }$ ${\displaystyle f\colon D\to \mathbb {R} }$ ${\displaystyle g\colon D\to \mathbb {R} }$ ${\displaystyle \lim _{x\to p}f(x)=a}$ ${\displaystyle \lim _{x\to p}g(x)=b}$ ${\displaystyle a,b\in \mathbb {R} }$ ${\displaystyle p}$ ${\displaystyle D}$ ${\displaystyle {\bar {\mathbb {R} }}=\mathbb {R} \cup \{-\infty ,+\infty \}}$

• ${\displaystyle \lim _{x\to p}(f(x)\pm g(x))=\lim _{x\to p}f(x)\pm \lim _{x\to p}g(x)=a\pm b}$
• ${\displaystyle \lim _{x\to p}(f(x)\cdot g(x))=\lim _{x\to p}f(x)\cdot \lim _{x\to p}g(x)=a\cdot b}$

Is additional , then also exists , and it holds ${\displaystyle b\neq 0}$ ${\displaystyle \lim _{x\to p}{\tfrac {f(x)}{g(x)}}}$

• ${\displaystyle \lim _{x\to p}{\frac {f(x)}{g(x)}}={\frac {\lim _{x\to p}f(x)}{\lim _{x\to p}g(x)}}={\frac {a}{b}}}$.

If both and apply, the limit value theorem cannot be applied. In many cases, however, the limit value can be determined using de l'Hospital's rule . ${\displaystyle \lim _{x\to p}f(x)=0}$ ${\displaystyle \lim _{x\to p}g(x)=0}$

• Nesting set :

Is and is , so is . ${\displaystyle |f(x)|\leq |g(x)|}$ ${\displaystyle \lim _{x\to p}g(x)=0}$ ${\displaystyle \lim _{x\to p}f(x)=0}$

• Chain rule :

From and with follows if it holds ( i.e. is continuous at the point ) or does not assume the value in a neighborhood of . Example: is wanted . The following applies to: ${\displaystyle \lim _{x\to p}f(x)=a}$ ${\displaystyle \lim _{u\to a}g(u)=L}$ ${\displaystyle a,L\in \mathbb {R} }$ ${\displaystyle \lim _{x\to p}g(f(x))=L}$ ${\displaystyle g(a)=L}$ ${\displaystyle g}$ ${\displaystyle a}$ ${\displaystyle f}$ ${\displaystyle p}$ ${\displaystyle a}$

${\displaystyle \lim _{x\to 0^{+}}\sin(x)^{\sin(x)}}$ ${\displaystyle 0<x<\pi }$

${\displaystyle \sin(x)^{\sin(x)}=e^{\sin(x)\cdot \ln(\sin(x))}}$

${\displaystyle f(x)=\sin(x)\cdot \ln(\sin(x))}$

${\displaystyle \lim _{x\to 0^{+}}\sin(x)\cdot \ln(\sin(x))=0}$ (According to de l'Hospital's rule)

Applying the chain rule with supplies ${\displaystyle g(u)=e^{u}}$

${\displaystyle \lim _{u\to 0}e^{u}=1\Rightarrow \lim _{x\to 0^{+}}\sin(x)^{\sin(x)}=1}$.

Application to the difference quotient

The application of the concept of limit value to difference quotients has proven to be particularly productive. It forms the real basis of analysis .

Differential quotient and differentiability Differential quotients (also called derivatives ) are the limit values ​​of the difference quotients of a function, i.e. expressions of the form

${\displaystyle \lim _{x_{1}\to x_{0}}{\frac {f(x_{1})-f(x_{0})}{x_{1}-x_{0}}}=\lim _{\Delta x\to 0}{\frac {\Delta y}{\Delta x}}}$

with and . Spellings are z. B. or , if this limit value exists. Differential calculus deals with the properties and the calculation of differential quotients . ${\displaystyle \Delta y:=f(x_{1})-f(x_{0})}$ ${\displaystyle \Delta x:=x_{1}-x_{0}}$ ${\displaystyle f'(x_{0})}$ ${\displaystyle {\frac {{\rm {d}}f}{{\rm {d}}x}}(x_{0})}$

If there is a differential quotient of a function at the point , then the function is called differentiable at the point . ${\displaystyle p}$ ${\displaystyle p}$

Important limit values

The in deriving the power functions with can limit occurring with the binomial theorem to calculate: ${\displaystyle f(x)=x^{n}}$ ${\displaystyle n\in \mathbb {N} }$

${\displaystyle {\frac {\mathrm {d} x^{n}}{\mathrm {d} x}}=\lim _{h\to 0}{\frac {(x+h)^{n}-x^{n}}{h}}=nx^{n-1}}$

The in deriving the exponential functions with the introduction of the required limit occurring Euler number and based thereon natural logarithm : ${\displaystyle f(x)=a^{x}}$ ${\displaystyle a\in \mathbb {R} ^{+}}$ ${\displaystyle e}$

${\displaystyle {\frac {\mathrm {d} a^{x}}{\mathrm {d} x}}=\lim _{h\to 0}{\frac {a^{x+h}-a^{x}}{h}}=a^{x}\lim _{h\to 0}{\frac {a^{h}-1}{h}}=a^{x}\ln a}$

The derivation of the trigonometric functions ultimately leads to the limit value . There are different approaches for calculating this limit value, depending on how the trigonometric functions and the number Pi are analytically defined. If you measure the angle in radians , you get ${\displaystyle \lim _{x\to 0}{\frac {\sin x}{x}}}$

${\displaystyle \lim _{x\to 0}{\frac {\sin x}{x}}=1.}$

Newer concept of limit value

More recently, a variant of the concept of limit values ​​has also been used that works with environments that are not dotted. Using sequences, this variant defines the limit value as follows: Let be a function, an element of the closed envelope and . Then we define: if and only if for every sequence with and where: . ${\displaystyle f\colon D\to \mathbb {R} }$ ${\displaystyle p}$ ${\displaystyle {\bar {D}}}$ ${\displaystyle L\in \mathbb {R} \cup \{\pm \infty \}}$ ${\displaystyle \lim _{x\to p}f(x)=L}$ ${\displaystyle (x_{n})_{n\in \mathbb {N} }}$ ${\displaystyle x_{n}\in D}$ ${\displaystyle \lim _{n\to \infty }x_{n}=p}$ ${\displaystyle \lim _{n\to \infty }f(x_{n})=L}$

The difference to the dotted variant given above is firstly that it is no longer prohibited if . Second, a definition on all points in the closed envelope is possible, in particular also on isolated points of . ${\displaystyle x_{n}=p}$ ${\displaystyle p\in D}$ ${\displaystyle {\bar {D}}}$ ${\displaystyle D}$

An equivalent non-dotted - definition of the limit value can also be easily given: In the above given - definition only needs to be replaced by , i.e. the case must also be expressly allowed. ${\displaystyle \varepsilon }$ ${\displaystyle \delta }$ ${\displaystyle \varepsilon }$ ${\displaystyle \delta }$ ${\displaystyle 0<|x-p|<\delta }$ ${\displaystyle |x-p|<\delta }$ ${\displaystyle x=p}$

The non-dotted version is not equivalent to the dotted version. It differs in particular at points of discontinuity:

In the dotted version is continuous in if and only if the limit of for exists and holds or if is an isolated point. In the non-dotted version, however, it is sufficient for continuity to require the existence of the limit value; the equation is thus automatically fulfilled. ${\displaystyle f}$ ${\displaystyle p\in D}$ ${\displaystyle f}$ ${\displaystyle x\to p}$ ${\displaystyle \lim _{x\to p}f(x)=f(p)}$ ${\displaystyle p}$ ${\displaystyle \lim _{x\to p}f(x)=f(p)}$

Example:

${\displaystyle f(x)={\begin{cases}0,&x\neq 0,\\1,&x=0.\end{cases}}}$

This function is not continuous. The limit value in the non-dotted sense does not exist. However, the limit value in the dotted sense does exist: because it is expressly required and applies to these values . It is obvious, however . ${\displaystyle \lim _{x\to 0}f(x)=0}$ ${\displaystyle x\neq 0}$ ${\displaystyle f(x)=0}$ ${\displaystyle \lim _{x\to 0}f(x)\neq f(0)}$

To avoid misunderstandings, the representatives of the non-dotted variant therefore recommend designating the dotted limit value of for as follows: ${\displaystyle f}$ ${\displaystyle x\to p}$

${\displaystyle \lim _{x\to p \atop x\neq p}f(x)}$

The representatives of the newer variant see the advantage of their variant compared to the classic dotted variant from Weierstraß in the fact that limit value sentences can be formulated more easily with the newer variant, because the special cases that result from the dotting no longer have to be taken into account.

Limit value of a function with respect to a filter

→ Main article : Filter convergence

Both the classic concept of limit value from Weierstrass and the newer limit value concept can be understood as special cases of the general limit value concept of a function with regard to a filter :

Let be a function from to , where is provided with a topology , and a filter on . One point is called the limit value of the function with respect to the filter when the filter produced by the filter base converges to, that is, when the filter produced by the filter base is finer than the ambient filter of . ${\displaystyle f}$ ${\displaystyle X}$ ${\displaystyle Y}$ ${\displaystyle Y}$ ${\displaystyle {\mathcal {F}}\subset {\mathcal {P}}(X)}$ ${\displaystyle X}$ ${\displaystyle L\in Y}$ ${\displaystyle f}$ ${\displaystyle {\mathcal {F}}}$ ${\displaystyle f\left({\mathcal {F}}\right)}$ ${\displaystyle L}$ ${\displaystyle f\left({\mathcal {F}}\right)}$ ${\displaystyle L}$

The newer definition for the limit value of a function in the point now corresponds to the special case that is selected as the environment filter of ; The classic definition by Weierstrass corresponds to the special case that the filter generated by the dotted surroundings of is chosen. ${\displaystyle x}$ ${\displaystyle {\mathcal {F}}}$ ${\displaystyle x}$ ${\displaystyle {\mathcal {F}}}$ ${\displaystyle x}$

See also

• Gaps in definition that can be continuously removed

Individual evidence

1. ↑ Harro Heuser: Textbook of Analysis. Part 1. 8th edition. BG Teubner, Stuttgart 1990, ISBN 3-519-12231-6 . Definition 38.1.
2. ↑ Harro Heuser: Textbook of Analysis. Part 2. 5th edition. BG Teubner, Stuttgart 1990, ISBN 3-519-42222-0 . Chapter 245 The new austerity. P. 697.
3. ↑ Harro Heuser: Textbook of Analysis. Part 1. 8th edition. BG Teubner, Stuttgart 1990, ISBN 3-519-12231-6 . Theorem 39.1.
4. ↑ Harro Heuser: Textbook of Analysis. Part 1. 8th edition. BG Teubner, Stuttgart 1990, ISBN 3-519-12231-6 . Definition 46.1.
5. ↑ Wikibooks: Archives of evidence: Analysis: Differential calculus: Differentiation of the sine function
6. ^ H. Amann, J. Escher: Analysis I. Birkhäuser, Basel 1998, ISBN 3-7643-5974-9 . P. 255.
7. ^ G. Wittstock: Lecture notes on Analysis 1st winter semester 2000–2001. Definition 2.3.27.
8. ↑ Harro Heuser: Textbook of Analysis. Part 1. 8th edition. BG Teubner, Stuttgart 1990, ISBN 3-519-12231-6 . Theorem 38.2.
9. ^ G. Wittstock: Lecture notes on Analysis 1st winter semester 2000-2001 . Comment 2.3.28, point 1.
10. ^ G. Wittstock: Lecture notes on Analysis 1st winter semester 2000-2001 Definition 2.3.2, Comment 3.
11. ^ G. Wittstock: Lecture notes on Analysis 1st winter semester 2000-2001. Comment 2.3.28 point 5.
12. ^ N. Bourbaki: Eléments de mathématique. Topology Générale. Springer, Berlin, ISBN 978-3-540-33936-6 . Chapitre I, § 7, Définition 3.
13. ^ N. Bourbaki: Eléments de mathématique. Topology Générale. Springer, Berlin, ISBN 978-3-540-33936-6 . Chapitre I, § 7.4.
14. ^ N. Bourbaki: Eléments de mathématique. Topology Générale. Springer, Berlin, ISBN 978-3-540-33936-6 . Chapitre I, § 7.5.

Web links

Commons : Limit of a function  - collection of images, videos and audio files