Squeeze theorem


In calculus, the squeeze theorem 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 [pi|], and was formulated in modern terms by Carl Friedrich Gauss.

Statement

The squeeze theorem is formally stated as follows.
  • The functions and are said to be lower and upper bounds 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.

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

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 close enough to 0. The correctness of which for positive can be seen by simple geometric reasoning that can be extended to negative as well. The second limit follows from the squeeze theorem and the fact that
for 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 and the arc on the unit circle has length . 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, and the height is 1. The area of the triangle is therefore
From the inequalities
we deduce that
provided , and the inequalities are reversed if . Since the first and third expressions approach as, and the middle expression approaches 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,