[Let's learn math #3] Sandwich Theorem and solving problems on it...

[amansharma555]

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Basic Introduction of Sandwich theorem

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Sandwich theorem, also called as Squeeze principle is an important theorem to find the limit of a function (which can't find directly or simple methods ) via comparing the function with two other functions whose limits are known or can be calculated easily.This theorem is very useful and also very important in field of Calculus and real analysis.

Sandwich theorem was used for the first time by mathematicians Archimedes and Eudoxus to compute the value of . But later on Carl Friedrich Gauss gave modern version of Sandwich theorem.

Sandwich theorem or Squeeze principle is also called as Pinching theorem, Sandwich rule, Squeeze lemma and two Policeman and a drunk theorem..

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Definition of Sandwich Theorem

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Let and be three functions such that:-

for all x near a or possibly at a see graph below:-

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And also it is given that:-

Then by Sandwich theorem:-

This theorem is also applicable in sequence, which is as follows:-

If

and , then

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Proof of Sandwich Theorem

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We have given that:-

-----(1)

Also

Then by using the definition of limit we have i.e given >0 ,however small there exists , belongs to set of real numbers such that :-

and

Let

= maximum of (,)

Then we have:-

and

Now by using the property of modulus we can say that:-

if ___(2)

and

if ___(3)

Therefore from (1),(2) and (3) we get:-

if

Which implies

if

Which implies

if

Therefore,

Hence Proved....

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How to use Sandwich theorem in functions

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In order to solve the various problems using Sandwich theorem, follow the following steps:-

• Write the given function...

For example,
Question:1)

(whose graph is top most), here we have to find the limit when x will approaches to 0

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• Then by using some property of given function, try to make two other functions whose limits are same as follows:-

For example,in this case the value of

is from [-1,1]

i.e.

Now multiply both sides by we get :-

Now as x approaches to zero, and - will become zero in other words, we can say that:-

, then by using the Sandwich theorem we can easily say that:-

, which is required solution..

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Lets do some more questions

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Question:2) Prove that:-

Solution:- Let

Now we know that value of Sinx always lies between [-1,1] , therefore we can say that:-

Dividing both sides by x we get:-

Now

Therefore, by Sandwich theorem:-

Hence proved....

Question 3):- Prove that:-

Solution:- Let

Now we know that value of Cosx lies between [-1,1] i.e. we have:-

On multiplying by -1 we get:-

or

Now on adding 2 we get :-

On dividing whole inequality by x+3 we get :-

Since,

Therefore, by Sandwich theorem:-

Hence proved....

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Applications of Sandwich Theorem

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• Sandwich theorem majorly help us in proving the existence and nature of limit of certain functions, which can not be find using common methods.

• It also help us to find whether the certain sequence is converges to a limits or not at a given point, by binding the sequence with two other sequences whose convergence is known to us at same point.

Citations:-

• (https://en.wikipedia.org/wiki/Squeeze_theorem) (Introduction,definition and proof of sandwich theorem and Question 1)

• (https://www.math.ucdavis.edu/~kouba/CalcOneDIRECTORY/squeezedirectory/SqueezePrinciple.html) Question 2 and 3 are taken from here.

• (https://www.quora.com/What-are-the-applications-of-sandwich-theorem) Applications of sandwich theorem has taken from here

• While all the graphs are made by me by using 3D geogebra and desmos. Also solutions to the problems are my own methods....

Thanks for Reading!