How to Use Squeeze Theorem

In calculus and mathematical analysis, the squeeze theorem is used to validate the limit of a function by comparing it to two other functions whose limits are known. It was initially utilised geometrically by the mathematicians Archimedes and Eudoxus in an attempt to compute, and Carl Friedrich Gauss defined it in contemporary terms.

Squeeze theorem or Pinch theorem are other names for the sandwich theorem. Using two functions in the neighbourhood, this is used to extrapolate the value of a function at a location x. The sandwich theorem is also related with an amusing incident.

What is Squeeze Theorem?

The squeeze theorem (also known as the sandwich theorem) asserts that if a function f(x) is sandwiched between two functions g(x) and h(x), and the limits of both g(x) and h(x) at a given position are equal (to L), then the limit of f(x) at that point is likewise equal to L. This appears to be similar to what we already know from algebra. If a = c and b = c, then b must be equal to c as well. According to the squeeze theorem, this rule also applies to limits.

The squeeze theorem is a theorem about the limit of a function caught between two other functions in calculus. It claims that two cops are accompanying a drunk. Both of these officers enter a cell, and despite the fact that the drunk is wobbling, he must also enter the cell. This is precisely how the theorem is used.

Limitations of Squeeze Theorem  

Squeeze (Sandwich) Theorem 

The Sandwich Theorem, also known as the squeeze theorem, is used to find the limits of trigonometric functions. The pinching theorem is another name for this proposition. The Sandwich theorem is generally used in math and fine analysis. This theorem is most probably used to determine a function’s limit by comparing it to two other functions whose limits are known or easily figured. 

In math and fine analysis, the squeeze theorem is used to confirm the limit of a function by comparing it to two other functions whose limits are known. It was first utilised geometrically by the mathematicians Archimedes and Eudoxus in an attempt to cipher, and Carl Friedrich Gauss defined it in ultramodern terms. 

This finding is also known as the sandwich theorem or sandwich rule in the United Kingdom. 

In that culture, the term “ sandwich” refers to confining food between two slices of chuck, as opposed to the more general term “ open sandwich,” which refers to a single slice of the chuck. 

 As a result, in colloquial British English, the ( occasionally unwelcome) circumstance of being squeezed between two realities is appertained to as being squeezed between them. 

The pinching theorem, also known as the sandwich theorem, the sandwich rule, the police theorem, the between theorem, and occasionally the squeeze lemma, is a mathematical theorem describing the limit of a function. The theorem is also known as the carabinieri theorem in Italy. 

Squeeze Theorem Statement

Suppose that f(x), g(x), and h(x) are three functions that is defined over an interval I such g(x) ≤ f(x) ≤ h(x) and suppose lim g(x) → a , lim h(x)  → a and lim h(x)→  L, then lim f(x) = L .

Because f is between g and h, these are f’s lower and upper bounds, respectively.

‘a’ does not have to be contained within I.

Application of Squeeze Theorem

For example, the limit of a function of two variables at a location (a, b) is the finite value L, such that the function approaches the unique value L for any disc centred at (a, b) whose radius tends to zero (thus the disc approaches its centre):

In polar coordinates, a disc is represented as:

As a result, a two-dimensional limit can be reduced to a one-dimensional limit.

The Squeeze Theorem is commonly used to determine the limit in polar coordinates.

Important Notes of Squeeze Theorem

Remember the following trig inequalities when utilising the squeeze theorem (or sandwich theorem) to evaluate a limit.

Cos x < (sin x)/x < 1

-1 ≤ sin x ≤ 1, for any x in the domain of sin x

-1 ≤ cos x ≤ 1, for any x in the domain of cos x

Conclusion

In limit calculus, the squeeze theorem is a crucial topic. It is used to determine a function’s limit. Sandwich Theorem, Pinching Theorem, Squeeze Lemma, and Sandwich Rule are all names for the Squeeze Theorem.

The Sandwich theorem is used to discover the limit of a function when it gets difficult or intricate, or when other methods have failed to identify it. The squeeze theorem is also known as the two officers theorem or a version thereof in several languages. According to the narrative, if two police officers are bringing an intoxicated prisoner between them to a cell, the prisoner must also end up in the cell.