Squeeze theorem
In calculus, the squeeze theorem (also known as the sandwich theorem, among other names) 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 , 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. 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. This theorem is also valid for sequences. Let formula_10 be two sequences converging to formula_11, and formula_12 a sequence. If formula_13 we have formula_14, then formula_12 also  converges to formula_11. Proof. According to the above hypotheses we have, taking the limit inferior and superior: formula_17 so all the inequalities are indeed equalities, and the thesis immediately follows. A direct proof, using the formula_18-definition of limit, would be to prove that for all real formula_19 there exists a real formula_20 such that for all formula_21 with formula_22, we have formula_23. Symbolically, formula_24 As formula_25 means that and formula_26 means that then we have formula_27 formula_28 We can choose formula_29. Then, if formula_22, combining () and (), we have formula_31 formula_32 which completes the proof. Q.E.D The proof for sequences  is very similar, using the formula_33-definition of the limit of a sequence. Examples. First example. The limit formula_34 cannot be determined through the limit law formula_35 because formula_36 does not exist. However, by the definition of the sine function, formula_37 It follows that formula_38 Since formula_39, by the squeeze theorem, formula_40 must also be 0. Second example. Probably the best-known examples of finding a limit by squeezing are the proofs of the equalities formula_41 The first limit follows by means of the squeeze theorem from the fact that formula_42 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 formula_43 for "x" close enough to 0. This can be derived by replacing formula_44 in the earlier fact by formula_45 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 formula_46 by squeezing, as follows. In the illustration at right, the area of the smaller of the two shaded sectors of the circle is formula_47 since the radius is sec "θ" and the arc on the unit circle has length Δ"θ". Similarly, the area of the larger of the two shaded sectors is formula_48 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 formula_49 From the inequalities formula_50 we deduce that formula_51 provided Δ"θ" > 0, and the inequalities are reversed if Δ"θ" < 0. Since the first and third expressions approach sec"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. formula_52 cannot be found by taking any number of limits along paths that pass through the point, but since formula_53 formula_54 formula_55 formula_56 formula_57 formula_58 therefore, by the squeeze theorem, formula_59