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 bounded 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.

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.

This theorem is also valid for sequences.

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 (ε, δ)-definition of limit, would be to prove that for all real ε > 0 there exists a real δ > 0 such that for all x with

Q.E.D The proof for sequences is very similar, using the

cannot be determined through the limit law

However, by the definition of the sine function,

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]

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

This can be derived by replacing sin x 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.

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.

Since the first and third expressions approach sec2θ as Δθ → 0, and the middle expression approaches

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