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#1 2014-10-29 03:34:26

Agnishom
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From: Riemann Sphere
Registered: 2011-01-29
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Formal Limits

Consider f(x) = x+3

How do I show that f(x) does not tend to 3 as x tends to 0, formally?


'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
I'm not crazy, my mother had me tested.

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#2 2014-10-29 04:12:06

Olinguito
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Re: Formal Limits


Bassaricyon neblina

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#3 2014-10-29 05:24:00

Agnishom
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From: Riemann Sphere
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Re: Formal Limits

I've tried that, what do you do about del?


'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
I'm not crazy, my mother had me tested.

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#4 2014-10-29 09:12:50

anonimnystefy
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From: Harlan's World
Registered: 2011-05-23
Posts: 16,049

Re: Formal Limits

How can you show something that is not correct?


“Here lies the reader who will never open this book. He is forever dead.
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#5 2014-10-29 11:12:05

bobbym
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From: Bumpkinland
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Posts: 109,606

Re: Formal Limits

You do have a point.


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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#6 2014-10-29 20:22:35

Bob
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Registered: 2010-06-20
Posts: 10,052

Re: Formal Limits

Thanks Stefy.  smile

I thought I was just being stupid.

Bob


Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob smile

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#7 2014-10-29 21:08:08

Agnishom
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From: Riemann Sphere
Registered: 2011-01-29
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Re: Formal Limits

Sorry guys sad

Let f(x) = x+3

How do I show that f(x) does not tend to 0 as x tends to 0, formally?


'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
I'm not crazy, my mother had me tested.

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#8 2014-10-30 01:42:47

Bob
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Registered: 2010-06-20
Posts: 10,052

Re: Formal Limits

Oh whaat.gif

You have to show that in a small neighbourhood of x = 0, f(x) is not in a small neighbourhood of zero.  As f(0) = 3, that shouldn't be hard.

Bob


Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob smile

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#9 2014-10-30 02:03:06

Agnishom
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From: Riemann Sphere
Registered: 2011-01-29
Posts: 24,974
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Re: Formal Limits

f(x) is not in a small neighbourhood of zero

What is a neighbourhood?


'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
I'm not crazy, my mother had me tested.

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#10 2014-10-30 02:13:11

Bob
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Registered: 2010-06-20
Posts: 10,052

Re: Formal Limits

If e is small then x +/-e is a neighbourhood of x.

And f(x +/-e) is the corresponding neighbourhood of f(x).

Let's take +e.  -e is similar.

Then f(0 + e) = 0 + e + 3

No matter how small e gets,  e + 3 is not close to 0.

http://en.wikipedia.org/wiki/Limit_(mathematics)

Bob


Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob smile

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#11 2014-10-30 02:17:49

Agnishom
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From: Riemann Sphere
Registered: 2011-01-29
Posts: 24,974
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Re: Formal Limits

But, I can make e big enough to make e+3 close to 0


'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
I'm not crazy, my mother had me tested.

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#12 2014-10-30 03:34:27

Bob
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Registered: 2010-06-20
Posts: 10,052

Re: Formal Limits

First you choose an e so that your x point is close to zero.  Then you generate f(0+e) .  If it's close to zero, then choose a smaller e.  The limit rule requires that you can get arbitrarily close.  Something like this:

   x           e             f(x+e)

   0           -3              0                 (oh dear I seem to have reached zero, but now to reduce e ..)
   0           -2              1
   0           -1              2
   0           -0.5           2.5
   0           -0.25         2.75
   0           -0.1           2.9
   0           -0.01         2.99

etc

As e gets smaller and I get closer to x = 0, f(x) does not get closer to zero. (It gets closer to 3 of course.)

So you might be able to choose an e that makes f(x) = 0, but that's not enough.  You have to show that as the x gets closer to zero, the f(x) gets closer to whatever.

Bob


Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob smile

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#13 2014-10-30 05:51:16

Agnishom
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From: Riemann Sphere
Registered: 2011-01-29
Posts: 24,974
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Re: Formal Limits

Oh, thanks!


'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
I'm not crazy, my mother had me tested.

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