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#1 2017-01-10 04:40:58

Zeeshan 01
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Registered: 2016-07-22
Posts: 648

Sandwich theorem

Can anyone please explain me sand which theorm ??


MZk

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#2 2017-01-10 08:54:47

Mathegocart
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Registered: 2012-04-29
Posts: 1,884

Re: Sandwich theorem

It is utilized such that limit f(x) is between two numbers.. and multiplied by another function to obtain another one.. we see the limit and see that f(x) is compressed into a number, which must be the limit
Graphically it looks quite elegant..
%28x%5E2%29sin%28x%5E%28-1%29%29.png


The integral of hope is reality.

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#3 2017-01-11 04:27:59

Zeeshan 01
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Registered: 2016-07-22
Posts: 648

Re: Sandwich theorem

Means ?


MZk

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#4 2017-01-11 05:14:27

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: Sandwich theorem

Roughly speaking if you know that your function (the blue line)stays between the green and the red line and you know that the limit at the point you are interested in for the red line is n and for the green line is n then the limit for your function at a is also n.


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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#5 2017-01-12 04:42:10

Zeeshan 01
Member
Registered: 2016-07-22
Posts: 648

Re: Sandwich theorem

??


MZk

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#6 2017-01-12 05:00:31

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: Sandwich theorem

What do you not understand?


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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#7 2017-01-12 06:56:18

zetafunc
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Registered: 2014-05-21
Posts: 2,130
Website

Re: Sandwich theorem

The standard example is with the function
We would like to understand how this function behaves as
approaches
Unfortunately, it's difficult to understand how
behaves for
tending to
We know that the inequality
always holds. But then if we multiply every term by
we obtain

and then taking the limit as
tends to 0, we see that
and the sandwich theorem tells us that, since its upper and lower bounds tend to the same limit, then so too must the function in the middle.

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#8 2017-01-12 15:23:55

Zeeshan 01
Member
Registered: 2016-07-22
Posts: 648

Re: Sandwich theorem

sandwich theorem tells us that, since its upper and lower bounds tend to the same limit, then so too must the function in the middle.


What is boundary and what is middle


MZk

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#9 2017-01-12 21:32:42

zetafunc
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Registered: 2014-05-21
Posts: 2,130
Website

Re: Sandwich theorem

The upper bound is the term on the right-hand side of the inequality, the lower bound is the term on the left-hand side. The middle is what goes between them.

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#10 2017-01-12 21:44:16

Zeeshan 01
Member
Registered: 2016-07-22
Posts: 648

Re: Sandwich theorem

So if it is sin○\○  eq 1


MZk

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#11 2017-01-12 22:19:38

zetafunc
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Registered: 2014-05-21
Posts: 2,130
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Re: Sandwich theorem

Yes, sin is bounded below by -1 and bounded above by 1.

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#12 2017-01-13 04:28:59

Zeeshan 01
Member
Registered: 2016-07-22
Posts: 648

Re: Sandwich theorem

Hmm!,


MZk

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#13 2017-01-13 04:29:59

Zeeshan 01
Member
Registered: 2016-07-22
Posts: 648

Re: Sandwich theorem

Yes, sin is bounded below by -1 and bounded above by 1.
So why it equals to 1???.


MZk

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#14 2017-01-13 05:57:38

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: Sandwich theorem

Since the limit of -x^2 as x approaches 0 is 0 and the limit of x^2 as x approaches 0 is 0 and

stays between them, the limit of it as x approaches 0 is also 0.

Intuitively we can understand by the following drawing that Mathegocart provided:

6MwTsXr.png

If your function the green line always stays between the red and the blue line, then as the red and the blue line get closer together the green line gets sandwiched in between.


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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