In calculus, the squeeze theorem, also known as the pinching theorem, the sandwich theorem, the sandwich rule, and sometimes the squeeze lemma, is a theorem regarding the limit of a function. In Italy, the theorem is also known as theorem of carabinieri. The squeeze theorem is used in calculus and mathematical analysis. It is typically used to confirm the limit of a function via comparison with two other functions whose limits are known or easily computed. It was first used geometrically by the mathematicians Archimedes and Eudoxus in an effort to compute pi|, and was formulated in modern terms by Carl Friedrich Gauss. In many languages, the squeeze theorem is also known as the two policemen theorem, or some variation thereof. The story is that if two policemen are escorting a drunk prisoner between them, and both officers go to a cell, then the prisoner must also end up in the cell.
Statement
The squeeze theorem is formally stated as follows.
Let I be an interval having the point a as a limit point. Let g, f, and h be functions defined on I, except possibly at a itself. Suppose that for every x in I not equal toa, we have and also suppose that Then
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.
Proof
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 and means that then we have We can choose. Then, if, combining and, we have which completes the proof. The proof for sequences is very similar, using the -definition of a limit of a sequence.
Statement for series
There is also the squeeze theorem for series, which can be stated as follows:
Let be two convergent series. Hence, the sequences are Cauchy. That is, for fixed, such that and similarly such that . We know that such that. Hence,, we have combining and : Therefore is a Cauchy sequence. So converges.
Examples
First example
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 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 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 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 − tan, 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 tan θ, the desired result follows.
Fourth example
The squeeze theorem can still be used in multivariable calculus but the lower must be below 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,
