Largest integer function !

mahmoudaljamel · 2006-10-28 20:55:35

f(x) = [ x ] + [ - x ]

proove that lim of f(x) as x approaches 2 , exicts and is not equal to f(2)

Dross · 2006-10-28 21:36:20

What do the square brackets mean? (as opposed to normal brackets)

I only ask because I haven't been told anything special to them, but they clearly mean it or the question is wrong/impossible.

mahmoudaljamel · 2006-10-29 03:39:05

Dross

the squar brackets stands for " largest integer number less than or equal x "
this is the notation used in the books, to refresh the memories of all members, these are examples : [1.5] = 1 , [4.3] = 4 , [-2.1] = -3

All_Is_Number · 2006-10-29 04:16:28

mahmoudaljamel wrote:

Dross
the squar brackets stands for " largest integer number less than or equal x "
this is the notation used in the books, to refresh the memories of all members, these are examples : [1.5] = 1 , [4.3] = 4 , [-2.1] = -3

Aha, the floor function.

So we have:

prove that lim of f(x) as x approaches 2 , exists and is not equal to f(2)

When x = 2:

When 1 < x < 2:

When 2 < x < 3:

However, I have been under the impression that the limit of f(x) as x approaches n is equal to f(n) if f(n) is defined. That seems to not be the case in this example, but I'm not positive. Perhaps a more knowledgeable poster will clarify.

Edit to add: This particular f(x) is not a continuous function, so we cannot assume that the limit of f(x) as x approaches n is equal to f(n).

Last edited by All_Is_Number (2006-10-29 04:26:52)

luca-deltodesco · 2006-10-29 04:25:40

i believe that actually number, the [] are meant to signify the nearest integer simply. rather than floor or ceiling.

All_Is_Number · 2006-10-29 04:30:14

luca-deltodesco wrote:

i believe that actually number, the [] are meant to signify the nearest integer simply. rather than floor or ceiling.

If that is the case, then the limit of f(x) as x approaches 2 is equal to f(2), so one cannot prove that the limit of f(x) as x approaches 2 is not equal to f(2).

Last edited by All_Is_Number (2006-10-29 04:31:55)

polylog · 2006-10-29 04:43:16

The definition given of [] here is exactly the definition of floor function: "largest integer number less than or equal x "; these must be floors.

If so then the Fourier series representation leads to an interesting way of looking at this:

Thus

Thus the limit exists, and is not equal to 2... !

This is probably not the way to do it, but it is an interesting result... can anyone see anything wrong with that?

The fourier series representation is correct I think, eg see here http://en.wikipedia.org/wiki/Floor_function.

All_Is_Number · 2006-10-29 04:45:55

polylog wrote:

Thus the limit exists, and is not equal to 0... !

There. I fixed it for ya! :D:D:P

polylog · 2006-10-29 04:47:32

Oh yeah, f(2). haha thanks

polylog · 2006-10-29 04:59:38

Ok here is another way, no Fourier:

We need to be aware of this formula:

therefore:

And so:

It is evident that this formula always gives -1.

Eg for 2.1 , the floor is 2, the ceiling is 3, and we are always taking the floor - ceiling, which is always going to be -1.

Thus the function simply reduces to f(x) = -1 for all x.

This makes the proof of the limit trivial.

krassi_holmz · 2006-10-29 12:43:28

polylog wrote:

Ok here is another way, no Fourier:
It is evident that this formula always gives -1.
Eg for 2.1 , the floor is 2, the ceiling is 3, and we are always taking the floor - ceiling, which is always going to be -1.
Thus the function simply reduces to f(x) = -1 for all x.
This makes the proof of the limit trivial.

Not so fast!
if x is integer, then

krassi_holmz · 2006-10-29 12:51:33

So we have:

Last edited by krassi_holmz (2006-10-29 13:01:37)

polylog · 2006-10-29 13:02:41

Oh right!

So the function actually reduces to:

Well we are trying to find

so let's restrict the values of x we are looking at to:

x is in that interval if it is approaching 2, and f(x) = -1 on that interval; it never reaches 2. So when evaluating the limit, f(x) = -1, and the limit is still -1 ... right?

krassi_holmz · 2006-10-29 13:07:28

Right!

mahmoudaljamel · 2006-10-30 03:59:01

Thanks for the very fruitful discussion; I would like to put the following notes:

1. The lim of f(x) as x approaches n is equal to f(n) if f(n) is defined , this statement should be changed to : if n is in the domain of f(x).

2. [x] = - [-x] if x is integer only

3. very interested to be introduced to the idea of flooring and ceiling , since I never see it in the American books

Ricky · 2006-10-30 07:07:50

1. The lim of f(x) as x approaches n is equal to f(n) if f(n) is defined , this statement should be changed to : if n is in the domain of f(x).

And f must be continuous.

Math Is Fun Forum

#1 2006-10-28 20:55:35

Largest integer function !

#2 2006-10-28 21:36:20

Re: Largest integer function !

#3 2006-10-29 03:39:05

Re: Largest integer function !

#4 2006-10-29 04:16:28

Re: Largest integer function !

#5 2006-10-29 04:25:40

Re: Largest integer function !

#6 2006-10-29 04:30:14

Re: Largest integer function !

#7 2006-10-29 04:43:16

Re: Largest integer function !

#8 2006-10-29 04:45:55

Re: Largest integer function !

#9 2006-10-29 04:47:32

Re: Largest integer function !

#10 2006-10-29 04:59:38

Re: Largest integer function !

#11 2006-10-29 12:43:28

Re: Largest integer function !

#12 2006-10-29 12:51:33

Re: Largest integer function !

#13 2006-10-29 13:02:41

Re: Largest integer function !

#14 2006-10-29 13:07:28

Re: Largest integer function !

#15 2006-10-30 03:59:01

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#16 2006-10-30 07:07:50

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