Squeeze theorem - Alchetron, The Free Social Encyclopedia

In calculus, the squeeze theorem, also known as the pinching theorem, the sandwich theorem, the sandwich rule, and sometimes the squeeze lemma, is a theorem regarding the limit of a function.

The squeeze theorem is used in calculus and mathematical analysis. It is typically used to confirm the limit of a function via comparison with two other functions whose limits are known or easily computed. It was first used geometrically by the mathematicians Archimedes and Eudoxus in an effort to compute π, and was formulated in modern terms by Gauss.

In many languages (e.g. French, German, Italian and Russian), the squeeze theorem is also known as the two policemen (and a drunk) theorem, or some variation thereof. The story is that if two policemen 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 policemen) the prisoner must also end up in the cell.

Statement

The squeeze theorem is formally stated as follows.

Let I be an interval having the point a as a limit point. Let f, g, 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:

g ( x ) ≤ f ( x ) ≤ h ( x )

and also suppose that:

lim x → a g ( x ) = lim x → a h ( x ) = L .

Then lim x → a f ( x ) = L .

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 → ∞ .

Proof

From the above hypotheses we have, taking the limit inferior and superior:

L = lim x → a g ( x ) ≤ lim inf x → a f ( x ) ≤ lim sup x → a f ( x ) ≤ lim x → a h ( x ) = L ,

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 | x − a | < δ , we have | f ( x ) − L | < ϵ . Symbolically,

∀ ϵ > 0 , ∃ δ > 0 : ∀ x , ( | x − a | < δ ⇒ | f ( x ) − L | < ϵ ) . .

As

lim x → a g ( x ) = L

means that

∀ ε > 0 , ∃ δ 1 > 0 : ∀ x ( | x − a | < δ 1 ⇒ | g ( x ) − L | < ε ) . ( 1 )

and

lim x → a h ( x ) = L

means that

∀ ε > 0 , ∃ δ 2 > 0 : ∀ x ( | x − a | < δ 2 ⇒ | h ( x ) − L | < ε ) , ( 2 )

then we have

g ( x ) ≤ f ( x ) ≤ h ( x ) g ( x ) − L ≤ f ( x ) − L ≤ h ( x ) − L

We can choose δ := min { δ 1 , δ 2 } . Then, if | x − a | < δ , combining (1) and (2), we have

− ε < g ( x ) − L ≤ f ( x ) − L ≤ h ( x ) − L < ε , − ε < f ( x ) − L < ε ,

which completes the proof. ◼

First example

The limit

lim x → 0 x 2 sin ⁡ ( 1 x )

cannot be determined through the limit law

lim x → a ( f ( x ) ⋅ g ( x ) ) = lim x → a f ( x ) ⋅ lim x → a g ( x ) ,

because

lim x → 0 sin ⁡ ( 1 x )

does not exist.

However, by the definition of the sine function,

− 1 ≤ sin ⁡ ( 1 x ) ≤ 1.

It follows that

− x 2 ≤ x 2 sin ⁡ ( 1 x ) ≤ x 2

Since lim x → 0 − x 2 = lim x → 0 x 2 = 0 , by the squeeze theorem, lim x → 0 x 2 sin ⁡ ( 1 x ) must also be 0.

Second example

Probably the best-known examples of finding a limit by squeezing are the proofs of the equalities

lim x → 0 sin ⁡ x x = 1 , lim x → 0 1 − cos ⁡ x x = 0.

The first follows by means of the squeeze theorem from the fact that

cos ⁡ x < sin ⁡ x x < 1

for x close enough, but not equal to 0.

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

d d θ tan ⁡ θ = sec 2 ⁡ θ

by squeezing, as follows.

In the illustration at right, the area of the smaller of the two shaded sectors of the circle is

sec 2 ⁡ θ Δ θ 2 ,

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

sec 2 ⁡ ( θ + Δ θ ) Δ θ 2 .

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. The area of the triangle is therefore

tan ⁡ ( θ + Δ θ ) − tan ⁡ ( θ ) 2 .

From the inequalities

sec 2 ⁡ θ Δ θ 2 ≤ tan ⁡ ( θ + Δ θ ) − tan ⁡ ( θ ) 2 ≤ sec 2 ⁡ ( θ + Δ θ ) Δ θ 2

we deduce that

sec 2 ⁡ θ ≤ tan ⁡ ( θ + Δ θ ) − tan ⁡ ( θ ) Δ θ ≤ sec 2 ⁡ ( θ + Δ θ ) ,

provided Δθ > 0, and the inequalities are reversed if Δθ < 0. Since the first and third expressions approach sec²θ as Δθ → 0, and the middle expression approaches (d/dθ) 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.

lim ( x , y ) → ( 0 , 0 ) x 2 y x 2 + y 2

cannot be found by taking any number of limits along paths that pass through the point, but since

0 ≤ x 2 x 2 + y 2 ≤ 1 − | y | ≤ y ≤ | y | − | y | ≤ x 2 y x 2 + y 2 ≤ | y | lim ( x , y ) → ( 0 , 0 ) − | y | = 0 lim ( x , y ) → ( 0 , 0 ) | y | = 0 0 ≤ lim ( x , y ) → ( 0 , 0 ) x 2 y x 2 + y 2 ≤ 0

therefore, by the squeeze theorem,

lim ( x , y ) → ( 0 , 0 ) x 2 y x 2 + y 2 = 0

References

Squeeze theorem Wikipedia

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