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 ...
, the squeeze theorem (also known as the sandwich theorem, among other names) is a
theorem
In mathematics, a theorem is a statement that has been proved, or can be proved. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical consequence of th ...
regarding the
limit
Limit or Limits may refer to:
Arts and media
* ''Limit'' (manga), a manga by Keiko Suenobu
* ''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 that is trapped between two other functions.
The squeeze theorem is used in calculus and
mathematical analysis
Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series (m ...
, typically to confirm the limit of a function via comparison with two other functions whose limits are known. It was first used geometrically by the
mathematician
A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems.
Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change.
History
On ...
s
Archimedes
Archimedes of Syracuse (;; ) was a Greek mathematician, physicist, engineer, astronomer, and inventor from the ancient city of Syracuse in Sicily. Although few details of his life are known, he is regarded as one of the leading scientists ...
and
Eudoxus in an effort to compute
, and was formulated in modern terms by
Carl Friedrich Gauss
Johann Carl Friedrich Gauss (; german: Gauß ; la, Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician and physicist who made significant contributions to many fields in mathematics and science. Sometimes refer ...
.
In many languages (e.g. French, German, Italian, Hungarian and Russian), the squeeze theorem is also known as the two officers (and a drunk) theorem, or some variation thereof. The story is that if two police officers are escorting a drunk prisoner between them, and both officers go to a cell, then (regardless of the path taken, and the fact that the prisoner may be wobbling about between the officers) the prisoner must also end up in the cell.
Statement
The squeeze theorem is formally stated as follows.
* The functions
and
are said to be
lower and upper bounds (respectively) of
.
* Here,
is ''not'' required to lie in the
interior
Interior may refer to:
Arts and media
* ''Interior'' (Degas) (also known as ''The Rape''), painting by Edgar Degas
* ''Interior'' (play), 1895 play by Belgian playwright Maurice Maeterlinck
* ''The Interior'' (novel), by Lisa See
* Interior de ...
of
. Indeed, if
is an endpoint of
, then the above limits are left- or right-hand limits.
* A similar statement holds for infinite intervals: for example, if
, then the conclusion holds, taking the limits as
.
This theorem is also valid for sequences. Let
be two sequences converging to
, and
a sequence. If
we have
, then
also converges to
.
Proof
According to the above hypotheses we have, taking the
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 ...
and superior:
so all the inequalities are indeed equalities, and the thesis immediately follows.
A direct proof, using the
-definition of limit, would be to prove that for all real
there exists a real
such that for all
with
, we have
. Symbolically,
As
means that
and
means that
then we have
We can choose
. Then, if
, combining () and (), we have
which completes the proof.
Q.E.D
Q.E.D. or QED is an initialism of the List of Latin phrases (full), Latin phrase , meaning "which was to be demonstrated". Literally it states "what was to be shown". Traditionally, the abbreviation is placed at the end of Mathematical proof#Endin ...
The proof for sequences is very similar, using the
-definition of the limit of a sequence.
Examples
First example
The limit
cannot be determined through the limit law
because
does not exist.
However, by the definition of the
sine function
In mathematics, sine and cosine are trigonometric functions of an angle. The sine and cosine of an acute angle are defined in the context of a right triangle: for the specified angle, its sine is the ratio of the length of the side that is opp ...
,
It follows that
Since
, by the squeeze theorem,
must also be 0.
Second example
Probably the best-known examples of finding a limit by squeezing are the proofs of the equalities
The first limit follows by means of the squeeze theorem from the fact that
for ''x'' close enough to 0. The correctness of which for positive x can be seen by simple geometric reasoning (see drawing) that can be extended to negative x as well. The second limit follows from the squeeze theorem and the fact that
for ''x'' close enough to 0. This can be derived by replacing
in the earlier fact by
and squaring the resulting inequality.
These two limits are used in proofs of the fact that the derivative of the sine function is the cosine function. That fact is relied on in other proofs of derivatives of trigonometric functions.
Third example
It is possible to show that
by squeezing, as follows.
In the illustration at right, the area of the smaller of the two shaded sectors of the circle is
since the radius is sec ''θ'' and the arc on the
unit circle
In mathematics, a unit circle is a circle of unit radius—that is, a radius of 1. Frequently, especially in trigonometry, the unit circle is the circle of radius 1 centered at the origin (0, 0) in the Cartesian coordinate system in the Eucl ...
has length Δ''θ''. Similarly, the area of the larger of the two shaded sectors is
What is squeezed between them is the triangle whose base is the vertical segment whose endpoints are the two dots. The length of the base of the triangle is tan(''θ'' + Δ''θ'') − tan(''θ''), and the height is 1. The area of the triangle is therefore
From the inequalities
we deduce that
provided Δ''θ'' > 0, and the inequalities are reversed if Δ''θ'' < 0. Since the first and third expressions approach sec
2''θ'' as Δ''θ'' → 0, and the middle expression approaches tan ''θ'', the desired result follows.
Fourth example
The squeeze theorem can still be used in multivariable calculus but the lower (and upper functions) must be below (and above) the target function not just along a path but around the entire neighborhood of the point of interest and it only works if the function really does have a limit there. It can, therefore, be used to prove that a function has a limit at a point, but it can never be used to prove that a function does not have a limit at a point.
cannot be found by taking any number of limits along paths that pass through the point, but since
therefore, by the squeeze theorem,
References
Notes
References
External links
*
Squeeze Theoremby Bruce Atwood (Beloit College) after work by, Selwyn Hollis (Armstrong Atlantic State University), the
Wolfram Demonstrations Project
The Wolfram Demonstrations Project is an organized, open-source collection of small (or medium-size) interactive programs called Demonstrations, which are meant to visually and interactively represent ideas from a range of fields. It is hos ...
.
Squeeze Theoremon ProofWiki.
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