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Sandwich Theorem in Mathematics.

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Sandwich theorem in mathematics.

sandwich-theorem-in-mathematics

Sandwich Theorem in Mathematics.

Outline of Sandwich Theorem:

1. Introduction

2. Statement

3. Proof

4. Examples

4.1: Example No.1
4.2: Example No.2
4.3: Example No.3

4.4: Example No.4

4.5: important Note

5. Applications of Sandwich Theorem.

6. References

1. Introduction:

A statement in mathematics or logic that can be rationally demonstrated to be true is called a theorem. Squeeze theorem is another name for the sandwich theorem. Basically, it is a theorem that deals with the limit of a function that is sandwiched between two other functions. To determine the limits of specific trigonometric functions, use the squeeze theorem or Sandwich Theorem.

The squeezing theorem is another name for this proposition. In calculus, especially mathematical analysis, the Sandwich theorem is generally applied. By comparing a limit of function to the limits of two other functions, whose limits are known, the squeeze theorem is commonly used in calculus and in mathematical analysis to verify a limit of function.

In an effort to compute, the mathematicians Archimedes and Eudoxus utilized it for the first time geometrically. Carl Friedrich Gauss then expressed it in more general terms. It is also called the Sandwich Theorem because it refers to a function ƒ whose values are sandwiched between the values of two other functions g and h that have the same limit L at a point c.


2. statement:

Suppose that g(x) ≤ f(x) ≤ h(x) for all x in some open interval containg c, except possibly at x = c itself, Suppose that lim x →c g(x) = lim x →c h(x) = L then lim x →c f(x) = L.

3. Proof:

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From the figure,

Area of ΔOAC < Area of sector OAC < Area of Δ OAB

⇒ 1 / 2 × OA × CD < x / 2π × (OA) 2< 1 / 2 × OA × AB

Here, OA is common so by cancelling the common terms, we get.

CD < x . OA < AB.

In triangle OCD,

sin x = CD/OC = CD/OA (since OC = OA as radii of same circle)

⇒ CD = OA sin x

Also,

tan x = AB/OA

⇒ AB = OA tan x

Thus,

OA sin x < OA . x < OA tan x

We know that length is always positive. So we can simplify the above inequality as:

Sin x < x < tan x …..(1)

From the given, 0 < x < π/2

In this interval, sin x is positive (in the first quadrant all trigonometric functions are positive).

Dividing eq. (1) by sin x,

(Sin x/sin x) < (x/sin x) < (tan x/sin x)

1 < (x/sin x) < (1/cos x) {since tan x = sin x/cos x}

Taking reciprocals throughout, we get.

Cos x < (sin x/x) < 1

Hence proved.


4. Examples of Sandwich Theorem:

4.1. Example No.1

Given that

1 - x2/4 ≤ u(x) ≤ 1+ x2/2

Find the lim x →0.

Solution:

Suppose that,

g(x) = 1 - x2/4 and h(x) = 1+ x2/2

Apply the lim x →0 on g(x) and h(x) it gives.

lim x →0 (1 - x2/4) = (1-0) = 1 and lim x →0 (11+ x2/2) = (1+0) = 1

So, according to sandwich theorem lim x →0 (u(x)) = 1

4.2. Example No.2

Given that:

lim θ →0 sin θ = 0

Solution:

As we know that the domain of sinθ is - | θ | ≤ Sin θ ≤ | θ |

Suppose that g(x) = - | θ | and h(x) = | θ |

Apply the lim θ→0 on g(x) and h(x) it gives,

lim θ→0 (- | θ |) = lim θ→0 (| θ |) = 0

So, according to sandwich theorem lim θ→0 Sinθ = 0

4.3. Example No. 3:

lim θ →0 Cosθ = 1

Solution:

As we know that the domain of 1- Cosθ is 0 ≤ 1 - Cos θ ≤ | θ |

Apply the lim θ→0 (1 - Cos θ) it gives,

lim θ→0 (1) - lim θ→0 (Cos θ )

= 1 – Cos (0)

= 1 – 0

= 1

So, according to sandwich theorem lim θ→0 Cos θ = 1

4. 4. Example No.4:

For and function f, lim x→c | f(x) | = 0 impels, lim x→c f(x) = 0

Solution:

Since,

-| f(x) | ≤ f(x) ≤ | f(x) | and -| f(x) | as well as | f(x) | have limit 0 as x→c.

So, by Sandwich theorem lim x→c f(x) = 0

4.5. Important Result:

Suppose that g(x) ≤ f(x) ≤ h(x) for all x in some open interval contains c, except possibly at x = c itself, and the limits of ƒ and g both exist as x approaches c, then

lim x→0 f(x) lim x→0 h(x)


5. Applications of Sandwich Theorem:

There are many applications of sandwich theorem in every field of science as well as in our daily life. Some discussed here,

  • Calculating the limits of given trigonometric functions involves applying the Sandwich Theorem, often known as the squeeze theorem.
  • It is primarily employed to demonstrate the existence and characteristics of the limit of some unusual functions.
  • Sandwich theorem is used to comparison of limit of one function to other functions.

https://hubpages.com/education/Limit-in-Mathematics

6. References:

Thomas, G.B., Weir, M.D., Hass, J., Heil, C. and Behn, A., 2010. Thomas' calculus: Early transcendentals (p. 510). Boston: Pearson.

This content is accurate and true to the best of the author’s knowledge and is not meant to substitute for formal and individualized advice from a qualified professional.

© 2022 Kinza Javaid

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