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#1 2013-01-06 09:45:52

anonimnystefy
Real Member
From: Harlan's World
Registered: 2011-05-23
Posts: 16,037

A weird limit!

Hi

Can anyone help me with the following limit:

I know that it should be equal to 1/2 and the the Stolz theorem is supposed to be used, but I cannot get the final result...

Here lies the reader who will never open this book. He is forever dead.
Taking a new step, uttering a new word, is what people fear most. ― Fyodor Dostoyevsky, Crime and Punishment
The knowledge of some things as a function of age is a delta function.

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#2 2013-01-06 10:12:31

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

Re: A weird 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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#3 2013-01-06 10:25:07

anonimnystefy
Real Member
From: Harlan's World
Registered: 2011-05-23
Posts: 16,037

Re: A weird limit!

No. I already looked at that page. It just gets the limit to the indeterminate from 0*infinity...

Here lies the reader who will never open this book. He is forever dead.
Taking a new step, uttering a new word, is what people fear most. ― Fyodor Dostoyevsky, Crime and Punishment
The knowledge of some things as a function of age is a delta function.

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#4 2013-01-06 10:51:59

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

Re: A weird limit!

We need to use the binomial theorem here now.

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 2013-01-06 10:54:32

anonimnystefy
Real Member
From: Harlan's World
Registered: 2011-05-23
Posts: 16,037

Re: A weird limit!

Do you see the n/(p+1) part? It makes it not possible to use Stolz like that.

Here lies the reader who will never open this book. He is forever dead.
Taking a new step, uttering a new word, is what people fear most. ― Fyodor Dostoyevsky, Crime and Punishment
The knowledge of some things as a function of age is a delta function.

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#6 2013-01-06 10:57:02

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

Re: A weird limit!

We are not allowed to split that limit?

Probably not, the two pieces are both equal to infinity.

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 2013-01-06 11:07:31

anonimnystefy
Real Member
From: Harlan's World
Registered: 2011-05-23
Posts: 16,037

Re: A weird limit!

That is right.

Here lies the reader who will never open this book. He is forever dead.
Taking a new step, uttering a new word, is what people fear most. ― Fyodor Dostoyevsky, Crime and Punishment
The knowledge of some things as a function of age is a delta function.

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#8 2013-01-06 11:14:58

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

Re: A weird limit!

I am trying to put it into the required form but the algebra is hideous.

The above limit equals 1 / 2 . But the limit is very difficult to handle.

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 2013-01-06 11:58:08

anonimnystefy
Real Member
From: Harlan's World
Registered: 2011-05-23
Posts: 16,037

Re: A weird limit!

I already got that much. That limit there is the problem...

Here lies the reader who will never open this book. He is forever dead.
Taking a new step, uttering a new word, is what people fear most. ― Fyodor Dostoyevsky, Crime and Punishment
The knowledge of some things as a function of age is a delta function.

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#10 2013-01-06 12:04:05

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

Re: A weird limit!

You got that far! How the heck did you get there?

Maybe we can use Stolz again?

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 2013-01-06 12:12:44

anonimnystefy
Real Member
From: Harlan's World
Registered: 2011-05-23
Posts: 16,037

Re: A weird limit!

It wasn't hard.

I don't think ot would get much prettier...

Last edited by anonimnystefy (2013-01-06 12:13:11)

Here lies the reader who will never open this book. He is forever dead.
Taking a new step, uttering a new word, is what people fear most. ― Fyodor Dostoyevsky, Crime and Punishment
The knowledge of some things as a function of age is a delta function.

Offline

#12 2013-01-06 12:42:20

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

Re: A weird limit!

Why do you think this is a job for Stoltz then?

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.

Offline

#13 2013-01-06 12:51:08

scientia
Member
Registered: 2009-11-13
Posts: 224

Re: A weird limit!

bobbym wrote:

The above limit equals 1 / 2 . But the limit is very difficult to handle.

Consider the numerator (ignoring the minus sign outside the fraction). Notice that the terms in
and
vanish so the highest power of
is
. Its coefficient is
. Now look at the denominator. The highest power of
is also
and its coefficient is
. Hence the limit as
is

using the following rule:

If

and

and
then

Last edited by scientia (2013-01-06 21:52:47)

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#14 2013-01-06 13:17:22

anonimnystefy
Real Member
From: Harlan's World
Registered: 2011-05-23
Posts: 16,037

Re: A weird limit!

Wow! That is amazing! Thank you!

Last edited by anonimnystefy (2013-01-06 13:17:45)

Here lies the reader who will never open this book. He is forever dead.
Taking a new step, uttering a new word, is what people fear most. ― Fyodor Dostoyevsky, Crime and Punishment
The knowledge of some things as a function of age is a delta function.

Offline

#15 2013-01-06 21:48:04

scientia
Member
Registered: 2009-11-13
Posts: 224

You're welcome.

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