Squeeze theorem

In calculus, the squeeze theorem (also known as the sandwich theorem, among other names[a]) is a theorem regarding the limit of a function that is trapped between two other functions.

Illustration of the squeeze theorem
When a sequence lies between two other converging sequences with the same limit, it also converges to this limit.

The squeeze theorem is used in calculus and mathematical analysis, 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 mathematicians Archimedes and Eudoxus in an effort to compute π, and was formulated in modern terms by Carl Friedrich Gauss.

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.[citation needed] 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.

StatementEdit

The squeeze theorem is formally stated as follows.[1]

Theorem —  Let I be an interval containing the point a. Let g, f, and h be functions defined on I, except possibly at a itself. Suppose that for every x in I not equal to a, we have

 
and also suppose that
 

Then  

  • The functions   and   are said to be lower and upper bounds (respectively) of  .
  • Here,   is not required to lie in the interior 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  .

ProofEdit

According to the above hypotheses we have, taking the limit inferior 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

 

 

 

 

 

(1)

and

 

means that

 

 

 

 

 

(2)

then we have

 
 

We can choose  . Then, if  , combining (1) and (2), we have

 
 

which completes the proof. Q.E.D

The proof for sequences is very similar, using the  -definition of the limit of a sequence.

ExamplesEdit

First exampleEdit

 
x2 sin(1/x) being squeezed in the limit as x goes to 0

The limit

 

cannot be determined through the limit law

 

because

 

does not exist.

However, by the definition of the sine function,

 

It follows that

 

Since  , by the squeeze theorem,   must also be 0.

Second exampleEdit

 
Comparing areas:
 

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[2]

 

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 exampleEdit

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 has length Δθ. 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 + Δθ) − tan(θ), and the height is 1. The area of the triangle is therefore

 

From the inequalities

 

we deduce that

 

provided Δθ > 0, and the inequalities are reversed if Δθ < 0. Since the first and third expressions approach sec2θ as Δθ → 0, and the middle expression approaches d/ tan θ, the desired result follows.

Fourth exampleEdit

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.[3]

 

cannot be found by taking any number of limits along paths that pass through the point, but since

 
 
 
 
 
 

therefore, by the squeeze theorem,

 

ReferencesEdit

NotesEdit

  1. ^ Also known as the pinching theorem, the sandwich rule, the police theorem, the between theorem and sometimes the squeeze lemma. In Italy, the theorem is also known as the theorem of carabinieri.

ReferencesEdit

  1. ^ Sohrab, Houshang H. (2003). Basic Real Analysis (2nd ed.). Birkhäuser. p. 104. ISBN 978-1-4939-1840-9.
  2. ^ Selim G. Krejn, V.N. Uschakowa: Vorstufe zur höheren Mathematik. Springer, 2013, ISBN 9783322986283, pp. 80-81 (German). See also Sal Khan: Proof: limit of (sin x)/x at x=0 (video, Khan Academy)
  3. ^ Stewart, James (2008). "Chapter 15.2 Limits and Continuity". Multivariable Calculus (6th ed.). pp. 909–910. ISBN 978-0495011637.

External linksEdit