Sandwich theorem

Zeeshan 01 · 2017-01-10 04:40:58

Can anyone please explain me sand which theorm ??

Mathegocart · 2017-01-10 08:54:47

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

Zeeshan 01 · 2017-01-11 04:27:59

Means ?

bobbym · 2017-01-11 05:14:27

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.

Zeeshan 01 · 2017-01-12 04:42:10

??

bobbym · 2017-01-12 05:00:31

What do you not understand?

zetafunc · 2017-01-12 06:56:18

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.

Zeeshan 01 · 2017-01-12 15:23:55

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

zetafunc · 2017-01-12 21:32:42

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.

Zeeshan 01 · 2017-01-12 21:44:16

So if it is sin○\○ eq 1

zetafunc · 2017-01-12 22:19:38

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

Zeeshan 01 · 2017-01-13 04:28:59

Hmm!,

Zeeshan 01 · 2017-01-13 04:29:59

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

bobbym · 2017-01-13 05:57:38

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:

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.

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

Sandwich theorem

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

Re: Sandwich theorem

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

Re: Sandwich theorem

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

Re: Sandwich theorem

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

Re: Sandwich theorem

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

Re: Sandwich theorem

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

Re: Sandwich theorem

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

Re: Sandwich theorem

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

Re: Sandwich theorem

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

Re: Sandwich theorem

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

Re: Sandwich theorem

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

Re: Sandwich theorem

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

Re: Sandwich theorem

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

Re: Sandwich theorem

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