Squeeze theorem

The Einschnürungssatz, Einschließungssatz, three- string set or sandwich set ( inter alia: Schachtelungssatz, respectively Quetschkriterium set of the two policemen, English sandwich theorem. ) Is in the Analysis a theorem on the limit of a function. According to the Einschnürungssatz seeks a function that is " squeezed " from above and below by two aspiring to the same value functions, even against this value.

The Einschnürungssatz is typically used to prove a limit of a function by one with two other compares the function known as their limits or are easy to determine. He was geometrically already used by mathematicians Archimedes and Eudoxus, to calculate the mathematical constant π. The modern formulation of the set originally by Carl Friedrich Gauss.

Formal Description

It is an interval that includes a value. There were, and to defined functions. If applies for each of

As well as

Then is.

Does not need to be surrounded by. If a boundary point of, then it is in the above limits to the left or right side. The same is true for infinite intervals: If, for example, the theorem also holds for the limit study.

The proof follows from the assumptions directly

So that the inequalities are actually equations and, thus, also sought.

Examples and Applications

The following examples show how the set is practically applied.

Example 1

Consider which is defined on all of except for. The limit for calculating the conventional way is hard: A direct substitution fails because the function is not defined at ( let alone continuous), and the rule of L' Hospital can not be applied because everywhere oscillates and no limit has. With matching upper and lower bounds functions, however, can apply the Einschnürungssatz.

Since the sine function is bounded by 1 in magnitude, magnitude is a suitable barrier for. In other words, the same with or:

And are polynomial functions and therefore continuous, therefore, applies

From the Einschnürungssatz now follows

Example 2

The above example is a particular application of a general case frequently occurring. Suppose we want to show that:

It is then sufficient to provide a function that is defined on an interval containing ( except possibly in ) to which,

And also applies to all of

Expressed in words, that the error between and can be made arbitrarily small, it selects close enough. These conditions are sufficient, since the absolute value function is not negative everywhere, so we

Can choose and apply the Einschnürungssatz. now

Applies and thus

Example 3

By elementary geometric considerations it can be shown that

Because of

Follows with the Einschnürungssatz

This threshold is helpful in determining the derivative of the sine function.

Evidence

The main idea of the proof is to consider the relative differences of the functions and. This has the effect that the lower bound function is constant zero, which makes the proof in detail much easier. The general case is then proved by algebraic means. In the special case and is

Be a fixed value. According to the definition of the limit of a function, there is now a so

For all of valid acceptance in accordance with

Therefore applies

It is concluded that

This proves that

For arbitrary and is now considered for each of

Now we subtracted from each term:

As is true of both therefore also strive against

With the proved above special case follows

And from this