Squeeze theorem

Squeeze theorem "Sandwich theorem" redirects here. For the result in measure theory, see Ham sandwich theorem. Illustration of the squeeze theorem When a sequence lies between two other converging sequences with the same limit, it also converges to this limit.

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.

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.

Contents 1 Statement 1.1 Proof 2 Examples 2.1 First example 2.2 Second example 2.3 Third example 2.4 Fourth example 3 References 3.1 Notes 3.2 References 4 External links Statement 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 {displaystyle g(x)leq f(x)leq h(x)} and also suppose that {displaystyle lim _{xto a}g(x)=lim _{xto a}h(x)=L.} Then {displaystyle lim _{xto a}f(x)=L.} The functions {textstyle g} and {textstyle h} are said to be lower and upper bounds (respectively) of {textstyle f} . Here, {textstyle a} is not required to lie in the interior of {textstyle I} . Indeed, if {textstyle a} is an endpoint of {textstyle I} , then the above limits are left- or right-hand limits. A similar statement holds for infinite intervals: for example, if {textstyle I=(0,infty )} , then the conclusion holds, taking the limits as {textstyle xto infty } .

This theorem is also valid for sequences. Let {displaystyle (a_{n}),(c_{n})} be two sequences converging to {displaystyle ell } , and {displaystyle (b_{n})} a sequence. If {displaystyle forall ngeq N,Nin mathbb {N} } we have {displaystyle a_{n}leq b_{n}leq c_{n}} , then {displaystyle (b_{n})} also converges to {displaystyle ell } .

Proof According to the above hypotheses we have, taking the limit inferior and superior: {displaystyle L=lim _{xto a}g(x)leq liminf _{xto a}f(x)leq limsup _{xto a}f(x)leq lim _{xto a}h(x)=L,} so all the inequalities are indeed equalities, and the thesis immediately follows.

A direct proof, using the {displaystyle (varepsilon ,delta )} -definition of limit, would be to prove that for all real {textstyle varepsilon >0} there exists a real {displaystyle delta >0} such that for all {displaystyle x} with {displaystyle |x-a|0,exists delta >0:forall x,(|x-a|0,exists delta _{1}>0:forall x (|x-a|0,exists delta _{2}>0:forall x (|x-a| 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 / dθ  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.[3] {displaystyle lim _{(x,y)to (0,0)}{frac {x^{2}y}{x^{2}+y^{2}}}} cannot be found by taking any number of limits along paths that pass through the point, but since {displaystyle 0leq {frac {x^{2}}{x^{2}+y^{2}}}leq 1} {displaystyle -left|yrightvert leq yleq left|yrightvert } {displaystyle -left|yrightvert leq {frac {x^{2}y}{x^{2}+y^{2}}}leq left|yrightvert } {displaystyle lim _{(x,y)to (0,0)}-left|yrightvert =0} {displaystyle lim _{(x,y)to (0,0)}left|yrightvert =0} {displaystyle 0leq lim _{(x,y)to (0,0)}{frac {x^{2}y}{x^{2}+y^{2}}}leq 0} therefore, by the squeeze theorem, {displaystyle lim _{(x,y)to (0,0)}{frac {x^{2}y}{x^{2}+y^{2}}}=0} References Notes ^ 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. References This article needs additional citations for verification. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed. Find sources: "Squeeze theorem" – news · newspapers · books · scholar · JSTOR (April 2010) (Learn how and when to remove this template message) ^ Sohrab, Houshang H. (2003). Basic Real Analysis (2nd ed.). Birkhäuser. p. 104. ISBN 978-1-4939-1840-9. ^ 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) ^ Stewart, James (2008). "Chapter 15.2 Limits and Continuity". Multivariable Calculus (6th ed.). pp. 909–910. ISBN 978-0495011637. External links Weisstein, Eric W. "Squeezing Theorem". MathWorld. Squeeze Theorem by Bruce Atwood (Beloit College) after work by, Selwyn Hollis (Armstrong Atlantic State University), the Wolfram Demonstrations Project. Squeeze Theorem on ProofWiki. Portal:  Mathematics Categories: Limits (mathematics)Functions and mappingsTheorems in calculusTheorems in real analysis