Discussion in 'Math' started by Anikets, Aug 10, 2014.

1. ### Anikets Thread Starter New Member

Aug 6, 2014
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what is concept of limit function?

2. ### studiot AAC Fanatic!

Nov 9, 2007
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Not sure what you mean by a limit function.

A limit, in mathematics, embodies the idea of a sequence of some sort getting ever closer to some fixed quantities.

So if the sequence is of numbers then if each term get closer and closer to some number then we say it approaches a limit

eg the sequence

{1/1, 1/2, 1/3, 1/4, 1/5......}

get closer and closer to zero the further along the sequence we go.

To have a limit function we would have to have a sequence of functions that became closer and closer to the particular series, the further we went along the sequence.

Fourier series provide and example of this, the more terms we employ the closer the Fourier series matches the function we are approximating.

Taylor / Maclaurin series are another example of this.

You will have to tell us more about the context if you want greater detail.

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3. ### socratus Active Member

Mar 26, 2012
267
3
. . . . .
Maybe mathematical limit functions are limited between
two fundamental physical limits :
a) infinite small: speed of quantum of light
b) infinite big: continuum T=0K
=

4. ### Wendy Moderator

Mar 24, 2008
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They can be. Speed of light for matter is a limit function, you can get as close as you have energy for but never hit it.

Fundamentally it is a basic calculus concept.

5. ### socratus Active Member

Mar 26, 2012
267
3
The physical borders for mathematical limits can be
quantum of light and zero continuum ( T=0K)
a)
" Speed of light for matter is a limit function,
you can get as close as you have energy for but never hit it."
/ Bill Marsden /
b)
All parameters of electron at interaction with vacuum became infinite.
==..

6. ### Wendy Moderator

Mar 24, 2008
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You do realize that made absolutely no sense? An electron is a form of matter, it has mass, there is bound by many of the properties of matter. We still use electrons in ballistic forms in many uses, tubes (aka valves), sputtering chambers, ionization of gases, etc.

7. ### socratus Active Member

Mar 26, 2012
267
3
1)
Renormalization
In quantum field theory, the statistical mechanics of fields,
and the theory of self-similar geometric structures,
renormalization is any of a collection of techniques used
to treat infinities arising in calculated quantities.
http://en.wikipedia.org/wiki/Renormalization
2)
" The shell game that we play to find n and j is technically
called renormalization. But no matter how clever the word,
it is what I would call a dippy process!
Having to resort to such hocus-pocus has prevented us
from proving the theory of quantum electrodynamics is
mathematically self-consistent. ....
I suspect that renormalization is not mathematically legitimate."
/ Book "QED - The Strange Theory of Light and Matter".
p.128. by Feynman /
==..

Mar 24, 2008
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9. ### socratus Active Member

Mar 26, 2012
267
3
OK. I will try to say my opinion in other way.
The constant speed of photon and zero vacuum are mathematical limits
because we cannot reach these parameters. (using physical instruments)
===.

10. ### MrAl Distinguished Member

Jun 17, 2014
3,618
760
Hi,

I like to restrict my discussions of limits to the purely mathematical because that is really the correct domain i think.

So in mathematics, a limit is a value of a function when one or more of that functions parameters approaches some set value from both sides.

For example, for the function:
y=1/x

when x approaches 4, y approaches 1/4, so the limit of f(x) as x approaches 4 is 0.25 .
To really evaluate this though we might consider x close to 4, and from both sides of 4 (slightly lower and slightly higher than 4). If we try values of x equal to 3.9, then 3.99, then 3.999, then 3.9999, etc., we find that as we let x get closer and closer to 4 (from the left) then y gets closer and closer to 1/4, and as we try values to the right such as 4.1, 4.01, 4.001, 4.0001, we find again that y gets closer and closer to 1/4, so the true limit is really 4.

If we allow x to approach positive infinity, then the fraction becomes very very small, and in fact reaches zero.

Similarly, the function:
y=1/(x+1)

for x approaching 4 again gives us a different result, namely:
y=1/5

so the limit now is y=1/5. But for x approaching infinity, that extra '1' in the denominator gets absorbed into x because x is so large now, so again we get:
y=0

as the limit as x approaches infinity. The reason for this is because infinity plus a constant is still infinity, so for any constant K:
y=1/(x+K)

the limit as x approaches infinity is always zero.

A little more tricky for example is:
y=x/(x+1)

Here, if we look for the limit as x approaches 4 again we get:
y=4/5

so the limit is 4/5. But if we let x approach infinity again, we get infinity over infinity which does not make too much sense by itself, so we have to take the derivative of the top and bottom:
(d[x]/dx)/(d[x+1]/dx)

and we are left with:
y=1/1

so the limit now as x approaches infinity equals just 1.
The above comes from a method known as L'Hospitals Rule where the direct evaluation leads to nonsense so we have to perform certain operations first to get the limit.

If the limit becomes infinite then it is said that there is no limit. For example,
y=1/x

as x approaches zero. In this case there is no limit although sometimes it is called 'infinity' which i believe is not correct. To see this, let x approach 0 from the left and x approach 0 from the right:
From the left: x=-0.1, x=-0.01, x=-0.001, x=-0.0001, etc., limit looks like -infinity.
From the right: x=0.1, x=0.01, x=0.001, x=0.0001, etc., limit looks like +infinity.
Since the right sided limit is not equal to the left sided limit it is said that there is no limit.

The formal definition is a bit more involved so you'll have to look further for more information. There's also one sided limits where we just consider what happens when x approaches a given value from just one side instead of both.

Note that we did not involve any physical process yet. When we do that the interpretation of the limit could get very application specific, meaning that sometimes a given interpretation may apply and sometimes not. That should probably be left for a study of physics.

Last edited: Aug 28, 2014
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11. ### studiot AAC Fanatic!

Nov 9, 2007
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No I don't agree.

For instance consider the function

$\left\{ y(x) = {x^2},\,x \not= 0 \\
y(x) = 1,\,x = 0 \\
\right\}$

The limit as the function approaches zero from both plus and minus is zero, but this does not equal the value of the function, since

$\lim_{x \to 0 - } = 0 = \lim_{x \to 0 + }$

What you have stated is the condition for the function to be continuous at a point (in this case the point zero).

That is a function is continuous at a point if the value of that function equals the limits when approached from + and from minus.

So my example function is not continuous at zero, nor is it differentiable at this point.
It is both at every other point, however.

Last edited: Aug 28, 2014

Jun 17, 2014
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760

Hello,

13. ### studiot AAC Fanatic!

Nov 9, 2007
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Mr Al, can you please tell me how you went about quoting since the quotes have not come out properly on my screen?

As the the function it was simply the function y = x squared, with the value of y=0 replaced by y=1 at the origin, but there are lots of possible examples.

I have started a discussion with the mods about the new format as I consider it a seriously retrograde step, since the most useful box of tricks on the right and the formatting tools appear to have vanished.

14. ### MrAl Distinguished Member

Jun 17, 2014
3,618
760
Hi,

Yes there is something wrong with the site software today. It looks like they 'upgraded' to that buggie Zenforo thing. That will mess things up for a long time to come.

So you mentioned y=x^2 right? What did you want to say about that function and limits?

15. ### studiot AAC Fanatic!

Nov 9, 2007
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519
AH, I see the Latex is now working

Thanks jpanhalt.

So Mr Al, please try post#12 again.

16. ### MrAl Distinguished Member

Jun 17, 2014
3,618
760
Hi again,

Ok so you have defined a function:
y=x^2 for x!=0,
y=1 for x=0

and stated that the limit appears to be 0 if we look at the left and right side limits, but it's 1 for x=0 so 0 cant be the limit.

But doesnt this function fail the, "Gotta be pretty dumb not to notice", theorem? (with reference to y=1 at x=0).

For a function that is defined at x that clearly we can certainly see that the limit can not be 0, it can not even exist at 0.
Also, you'll note that there is a more formal definition which i didnt want to get into here.

17. ### studiot AAC Fanatic!

Nov 9, 2007
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519
I'm sorry I don't understand this response at all.

The perfectly respectable mathematical function I suggested (because it is easy to use as an example not because it is common) has perfectly respectable limits (from above and below) at x=0.
It also has a perfectly respectable value that is not equal to this limit.

This contradicts your statement that such limits are defined as the value of the function.

They are not.

The value of a function and its limits are different mathematical entities.
They may coincide.
They may not.

When they do not the function is discontinuous.
When they coincide the function is continuous.

There are worse behaviours such as a square wave,

or the modulus function which is continuous at x=0 but not differentiable at this point.

Last edited: Aug 29, 2014
18. ### MrAl Distinguished Member

Jun 17, 2014
3,618
760
Hello again,

I think i see what you are trying to get at, but i did already agree that there are "one sided" limits. That means that the function may have two different limits depending on which direction we approach from. We can still call them limits, but then it's not a two sided limit.
Since it's been a long time since i had to look at this i'll take another look and try to get back here with more info.

19. ### studiot AAC Fanatic!

Nov 9, 2007
5,005
519
Hello again MrAl.

Whilst you are looking again, have another look at my post#2

The limit here is not one sided as it can only be approached one way.
Further the limit is not a member of the set ie it is not part of the sequence since there is no n for which 1/n = 0

Wish I could do a smiley here

20. ### Wendy Moderator

Mar 24, 2008
20,993
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Try colon and a parenthesis.