Constriction rate

Sandwich theorem: If a sequence lies between two converging sequences with the same limit value, then it must also converge towards this limit value.

The constriction clause, inclusion clause, three-sequence clause or sandwich clause (among others: nesting clause, squeeze lamb or clause about the two policemen; English sandwich theorem ) is a clause in analysis about the limit value of a function . According to the constriction principle, a function that is “constrained” from above and below by two functions tending towards the same value also tends towards this value.

The constriction theorem is typically used to detect a limit of a function by comparing the function to two others whose limits are known or easily determined. It was already used geometrically by the mathematicians Archimedes and Eudoxus to calculate the circle number π . The modern formulation of the sentence originally comes from Carl Friedrich Gauß .

The theorem applies in particular to limit values ​​of sequences: a function that is bounded from above and below by two sequences tending towards the same value also converges towards this value.

Containment rule for episodes

Be and two real consequences with , and for almost everyone (all but a finite number) . If another episode is with for almost everyone , then it converges , also against . ${\displaystyle x_{n}}$ ${\displaystyle y_{n}}$ ${\displaystyle x_{n}\rightarrow a}$ ${\displaystyle y_{n}\rightarrow a}$ ${\displaystyle x_{n}\leq y_{n}}$ ${\displaystyle n}$ ${\displaystyle w_{n}}$ ${\displaystyle x_{n}\leq w_{n}\leq y_{n}}$ ${\displaystyle n}$ ${\displaystyle w_{n}}$ ${\displaystyle a}$

example

Be

${\displaystyle w_{n}={\frac {1}{n+\log {n}^{2}+3}}}$

one episode. As for the denominator is always greater than . Therefore applies ${\displaystyle \log {n}\geq 0}$ ${\displaystyle n\geq 1}$ ${\displaystyle n}$

${\displaystyle 0\leq w_{n}\leq {\frac {1}{n}}}$.

Since both and converge against , it follows from the inclusion rule that also converges against . ${\displaystyle {\frac {1}{n}}}$ ${\displaystyle 0}$ ${\displaystyle 0}$ ${\displaystyle w_{n}}$ ${\displaystyle 0}$

Constriction set for functions

Let it be an interval that contains a value . Let there be , and on defined functions. If for each of true ${\displaystyle I}$ ${\displaystyle a}$ ${\displaystyle f}$ ${\displaystyle g}$ ${\displaystyle h}$ ${\displaystyle I\setminus \lbrace a\rbrace }$ ${\displaystyle x\neq a}$ ${\displaystyle I}$

${\displaystyle g(x)\leq f(x)\leq h(x),}$

such as

${\displaystyle \lim _{x\to a}g(x)=\lim _{x\to a}h(x)=L}$,

then is . ${\displaystyle \lim _{x\to a}f(x)=L}$

${\displaystyle a}$does not have to lie in the middle of . If the edge point is of , the above limit values ​​are left or right-hand side. The same applies to infinite intervals: if, for example , the sentence also applies to the limit value analysis . ${\displaystyle I}$ ${\displaystyle a}$ ${\displaystyle I}$ ${\displaystyle I=[0,\infty )}$ ${\displaystyle x\to \infty }$

As a proof it follows directly from the assumptions

${\displaystyle L=\lim _{x\to a}g(x)\leq \liminf _{x\to a}f(x)\leq \limsup _{x\to a}f(x)\leq \lim _{x\to a}h(x)=L}$,

so that the inequalities are actually equations and therefore also strive against . ${\displaystyle f}$ ${\displaystyle L}$

Examples and Applications

The following examples show how the theorem is applied in practice.

example 1

f (blue) with barrier functions g (red) and h (green)

Consider that it is defined on all but for . It is difficult to calculate the limit value for in the conventional way: A direct substitution fails because the function at is not defined (let alone continuous ), and de l'Hospital's rule cannot be applied either, since it oscillates everywhere and there is no limit value Has. However, the constriction theorem can be used with suitable upper and lower limit functions. ${\displaystyle f(x)=x^{2}\sin \!\left({\tfrac {1}{x}}\right)}$ ${\displaystyle \mathbb {R} }$ ${\displaystyle x=0}$ ${\displaystyle x\to 0}$ ${\displaystyle x=0}$ ${\displaystyle \sin \!\left({\tfrac {1}{x}}\right)}$

Since the amount of the sine function is limited by 1, there is a suitable limit for . In other words with and : ${\displaystyle x^{2}}$ ${\displaystyle f}$ ${\displaystyle g(x)=-x^{2}}$ ${\displaystyle h(x)=x^{2}}$

${\displaystyle {\begin{matrix}-1&\leq &\sin {\frac {1}{x}}&\leq &1\\-x^{2}&\leq &x^{2}\sin {\frac {1}{x}}&\leq &x^{2}\\g(x)&\leq &f(x)&\leq &h(x)\end{matrix}}}$

${\displaystyle g}$and are polynomial functions and are therefore continuous, therefore ${\displaystyle h}$

${\displaystyle \lim _{x\to 0}g(x)=\lim _{x\to 0}h(x)=0}$.

From the constriction theorem it now follows

${\displaystyle \lim _{x\to 0}f(x)=0}$.

Example 2

The above example is a special application of a common general case. Suppose we want to show that:

${\displaystyle \lim _{x\to a}f(x)=L}$.

It is then sufficient to find a function that is defined on a containing interval (except possibly at ) for which holds ${\displaystyle h}$ ${\displaystyle a}$ ${\displaystyle I}$ ${\displaystyle a}$

${\displaystyle \lim _{x\to a}h(x)=0}$,

and also for all of true ${\displaystyle x\neq a}$ ${\displaystyle I}$

${\displaystyle |f(x)-L|\leq h(x)}$.

In words, this means that the error between and can be made as small as you want if you choose close enough . These conditions are sufficient since the magnitude function is not negative everywhere, so that we ${\displaystyle f(x)}$ ${\displaystyle L}$ ${\displaystyle x}$ ${\displaystyle a}$

${\displaystyle g(x)=0}$ for all ${\displaystyle x}$

can choose and apply the constriction rate. Because now

for true , ${\displaystyle x\to a}$ ${\displaystyle |f(x)-L|\to 0}$

also applies and thus ${\displaystyle f(x)-L\to 0}$

${\displaystyle f(x)=(f(x)-L)+L\longrightarrow 0+L=L}$.

Example 3

${\displaystyle {\begin{aligned}&\,F(\triangle ADF)\geq F({\text{Sektor}}\,ADB)\geq F(\triangle ADB)\\\Rightarrow &\,{\frac {1}{2}}\cdot \tan(x)\cdot 1\geq {\frac {x}{2\pi }}\cdot \pi \geq {\frac {1}{2}}\cdot \sin(x)\cdot 1\\\Rightarrow &\,{\frac {\sin(x)}{\cos(x)}}\geq x\geq \sin(x)\\\Rightarrow &\,{\frac {\cos(x)}{\sin(x)}}\leq {\frac {1}{x}}\leq {\frac {1}{\sin(x)}}\\\Rightarrow &\,\cos(x)\leq {\frac {\sin(x)}{x}}\leq 1\end{aligned}}}$

Elementary geometric considerations on the unit circle (see drawing on the right) show that

${\displaystyle \cos(x)\leq {\frac {\sin(x)}{x}}\leq 1}$.

Because of

${\displaystyle \lim _{x\to 0}\cos(x)=1}$

follows with the constriction theorem

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

This limit value helps determine the derivative function of the sine.

proof

The main idea of this evidence is to the relative differences in functions , and to look at. This has the effect that the lower bound function is constantly zero, which makes the proof much easier in detail. The general case is then proved in an algebraic way. In the special case and applies ${\displaystyle f}$ ${\displaystyle g}$ ${\displaystyle h}$ ${\displaystyle g(x)=0}$ ${\displaystyle L=0}$

${\displaystyle \lim _{x\to a}h(x)=0}$.

Be a fixed value. According to the definition of the limit of a function, there is now such that ${\displaystyle \varepsilon >0}$ ${\displaystyle \delta >0}$

if holds , then is . ${\displaystyle 0<|x-a|<\delta }$ ${\displaystyle |h(x)|<\varepsilon }$

For all off applies according to assumption ${\displaystyle x\neq a}$ ${\displaystyle I}$

${\displaystyle 0=g(x)\leq f(x)\leq h(x)}$,

therefore applies

${\displaystyle |f(x)|\leq |h(x)|}$.

From this one concludes that

if holds , then is . ${\displaystyle 0<|x-a|<\delta }$ ${\displaystyle |f(x)|\leq |h(x)|<\varepsilon }$

This proves that

${\displaystyle \lim _{x\to a}f(x)=0}$.

For any and now applies to each of ${\displaystyle g}$ ${\displaystyle L}$ ${\displaystyle x\neq a}$ ${\displaystyle I}$

${\displaystyle g(x)\leq f(x)\leq h(x)}$.

Now one subtracts from each expression: ${\displaystyle g(x)}$

${\displaystyle 0\leq f(x)-g(x)\leq h(x)-g(x)}$.

Since it applies to striving both for and against ${\displaystyle x\to a}$ ${\displaystyle g(x)}$ ${\displaystyle h(x)}$ ${\displaystyle L}$

${\displaystyle h(x)-g(x)\to L-L=0}$.

With the special case proved above, it follows

${\displaystyle f(x)-g(x)\to 0}$ For ${\displaystyle x\to a}$

and then from it

${\displaystyle f(x)=(f(x)-g(x))+g(x)\to 0+L=L}$.

Generalizations

A measure-theoretical generalization is Pratt's theorem , in which, through the constriction by means of locally convergent function sequences, conclusions can be drawn about the interchangeability of limit value formation and integration of the constricted function sequence as well as the integrability of the limit function.

literature

• Wolfgang Walter : Analysis 1 . Springer, 5th edition, 2013, ISBN 9783662056981 , pp. 63, 119

Web links

Wikibooks: Math for Non-Freaks: Sandwich Set, Constriction Set, Containment Set  - Study and Teaching Materials

• C. Sean Bohun: The Squeeze Theorem. ( PDF ) February 9, 2013, accessed April 6, 2013 . 
• Joseph M. Ling: Examples on Limits of Functions: The Squeeze Theorem. (PDF; 96 kB) October 2, 2001, archived from the original on September 20, 2008 ; accessed on February 9, 2013 . 

Individual evidence

1. ↑ Harro Heuser : Textbook of Analysis . Math guides. 17th edition. Part 1. Vieweg + Teubner (Springer), Wiesbaden 2009, ISBN 978-3-8348-0777-9 , p. 152 ( excerpt (Google) ).
2. ↑ Selim G. Krejn, VN Uschakowa: preliminary stage to higher mathematics . Springer, 2013, ISBN 9783322986283 , pp. 80-81 . See also Salman Khan  : Proof: limit of (sin x) / x at x = 0 (video, Khan Academy (English))