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#1 2014-10-23 00:56:52

TeoLeo
Member
Registered: 2014-10-23
Posts: 3

A question about limits

Ok guys, I can't work out this sequence limit:

lim  sin( π√(4n^2+n) ) = ?
n⇒∞

I'm sure it's not really that hard, but...help, please!! big_smile

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#2 2014-10-23 03:20:29

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

Re: A question about limits

Hi;

I would say that limit does not exist.


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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#3 2014-10-23 03:20:59

Olinguito
Member
Registered: 2014-08-12
Posts: 649

Re: A question about limits

This is a tricky one. Consider

[list=*]
[*]

[/*]
[/list]

Hence

[list=*]
[*]

[/*]
[/list]

Last edited by Olinguito (2014-10-23 03:22:28)


Bassaricyon neblina

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#4 2014-10-23 03:56:33

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

Re: A question about limits

Hi;

Are you sure you are allowed to take the limit inside the sin() like that?


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 2014-10-23 04:11:24

Olinguito
Member
Registered: 2014-08-12
Posts: 649

Re: A question about limits

For continuous functions f, yes:

[list=*]
[*]

[/*]
[/list]

In fact this is part of the definition of continuousness!


Bassaricyon neblina

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#6 2014-10-23 04:17:09

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

Re: A question about limits

Hi;

But this is not a continuous function. It says a sequence limit, implying a discrete function. I am not trying to be a nit, the experts agree with your answer. I am just trying to see if I can understand the question like you do. Thanks.


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 2014-10-23 04:32:11

Olinguito
Member
Registered: 2014-08-12
Posts: 649

Re: A question about limits

bobbym wrote:

But this is not a continuous function.

Of course the it is! It’s the sine function.

Wikipedia wrote:

For any continuous function f, if
then
.

http://en.wikipedia.org/wiki/Limit_of_a … Properties

Last edited by Olinguito (2014-10-23 04:35:31)


Bassaricyon neblina

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#8 2014-10-23 04:33:48

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

Re: A question about limits

Hi;

Okay, thanks for the help.


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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#9 2014-10-23 04:57:30

Olinguito
Member
Registered: 2014-08-12
Posts: 649

Re: A question about limits

Here’s what I did on Excel:

[list=*]
[*]SinPiOver4.png[/*]
[/list]


Bassaricyon neblina

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#10 2014-10-23 05:01:46

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

Re: A question about limits

Hi;

Yes, that is true but notice that you also made the decision to treat n as an integer. If you did not then the limit would not exist.


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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#11 2014-10-23 05:10:40

Olinguito
Member
Registered: 2014-08-12
Posts: 649

Re: A question about limits

bobbym wrote:

Hi;

Yes, that is true but notice that you also made the decision to treat n as an integer. If you did not then the limit would not exist.

I agree.

[list=*]
[*]

[/*]
[/list]

definitely does not exist. But

[list=*]
[*]

[/*]
[/list]

does – since in this case we ignore all values that are not the image of integer n.


Bassaricyon neblina

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#12 2014-10-23 05:11:41

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

Re: A question about limits

So why choose n as an integer?


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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#13 2014-10-23 05:14:50

Olinguito
Member
Registered: 2014-08-12
Posts: 649

Re: A question about limits

Why not? We’re dealing with sequences here, aren’t we?

Last edited by Olinguito (2014-10-23 05:18:26)


Bassaricyon neblina

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#14 2014-10-23 05:21:14

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

Re: A question about limits

That is the answer. I did not see that and assumed a function limit.


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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#15 2014-10-23 05:28:12

Olinguito
Member
Registered: 2014-08-12
Posts: 649

Re: A question about limits

Let’s see what the OP says.


Bassaricyon neblina

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#16 2014-10-23 07:28:12

TeoLeo
Member
Registered: 2014-10-23
Posts: 3

Re: A question about limits

Hi, Olinguito.
I don't really understand the first part with "Consider: √(4n^2+n) - 2n" .

What's with that 2n?

understand everything else after that...so is "-2n" added there just randomly to help us? Or why did you chooose it?

Last edited by TeoLeo (2014-10-23 07:45:06)

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#17 2014-10-23 20:36:29

Olinguito
Member
Registered: 2014-08-12
Posts: 649

Re: A question about limits

Notice that

[list=*]
[*]

[/*]
[/list]

That’s pretty useless to us. What we need is more than that: we need to show

[list=*]
[*]

[/*]
[/list]

In order to do that we show

[list=*]
[*]

[/*]
[/list]

Last edited by Olinguito (2014-10-23 20:42:43)


Bassaricyon neblina

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