Squeeze Theorem – Limits
Squeeze theorem is also known as the sandwich theorem, the sandwich rule, the police theorem, the pinching theorem sometimes the squeeze lemma, the theorem of carabinieri is used in mathematical analysis to find the limit of a function when there are two other functions whose limits are known are present. The squeeze theorem is applied in calculus and mathematical analysis. It is typically applied 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 Carl Friedrich Gauss.
The squeeze theorem is formally stated as follows:
Let functions f(x), g(x), h(x) be real functions such that h(x) ≤ g(x) ≤ f(x). Then for all x in the domain of the function,
Here the functions h and f are said to be lower and upper bounds of g respectively. The above function is represented in a graph below.
Consider the following image:
In the diagram above O is the centre of the circle AB is a tangent of the circle from point B and CD is perpendicular to OB and angle COD is x radians
By taking the reciprocals we get
From the above inequality, we proved that
From this equation, we understand that (sin x/x) always lies between cos x and 1. So (sin x/x) is sandwiched between 1 and cos x. We know that
To prove this limit we use the trigonometric identity