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Concept
Squeeze theorem
Summary
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 pi, and was formulated in modern terms by Carl Friedrich Gauss.
The squeeze theorem is formally stated as follows.
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
and also suppose that
Then
The functions g and h are said to be lower and upper bounds (respectively) of f.
Here, a is not required to lie in the interior of I. Indeed, if a is an endpoint of I, then the above limits are left- or right-hand limits.
A similar statement holds for infinite intervals: for example, if I = (0, ∞), then the conclusion holds, taking the limits as x → ∞.
This theorem is also valid for sequences. Let (a_n), (c_n) be two sequences converging to ℓ, and (b_n) a sequence. If we have a_n ≤ b_n ≤ c_n, then (b_n) also converges to ℓ.
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 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.
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.
Official source
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