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April 15th, 2012, 07:35 AM   #1
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calculus doubts

Hi guys,
please help me with the following questions.

1) lim x-->0 x[1/x] where [.] denotes the greatest integer function.

2) if f "(x) exists, then show that lim h-->0 {f(x+h)-2f(x)+f(x-h)}/h^2 = f "(x)

3) lim x-->0 [x]/x (for this one i'm getting 0 as the RHL and -oo as the RHL. please confirm!)
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April 15th, 2012, 07:54 AM   #2
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Re: calculus doubts

1. If x is integer, then x/[x] lim x -> 0 is 0
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April 15th, 2012, 08:18 AM   #3
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Re: calculus doubts

2. f''(x) = d(dy) / dx^2.
d(dy) = d(f(x+h)-f(x)) = d(f(x+h)) - d(f(x)) = f(x+2h) - f(x) - f(x+h)-f(x)
Now set x+h = t
Then f(t+h)-2f(x)-f(t) = d(dy)
As h tense to zero, x tense to t.
Thus it is proved
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April 15th, 2012, 12:20 PM   #4
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Re: calculus doubts

Quote:
Originally Posted by mathbalarka
1. If x is integer, then x/[x] lim x -> 0 is 0
I think that this is not true. We have where (with I indicate the fractional part); and we also have that .

So
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April 15th, 2012, 12:24 PM   #5
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Re: calculus doubts

You can use a quite similar argument to solve the third question...
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April 15th, 2012, 08:48 PM   #6
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Re: calculus doubts

Thanks delirium. But what argument would you use for the last question? I got this:
lim x-->0 x-{x} whole divided by x
Now what should I do? If x is very close to zero, then x={x} isn't it?
Why can't I do the following:
For x-->0+, [x]=0. So limit =0. For x-->0-, [x]=-1 so the limit is -oo ??
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April 16th, 2012, 04:38 AM   #7
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Re: calculus doubts

Your limit is: ; but is defined as the biggest integer smaller than ; so, if , is steadly .

But if , so I think that this limit doesn't exist, but I'm not sure.

EDIT: I found a confirmation in a www.mat.uniroma2.it/~tauraso/Online1/Lezioni/Lezioni1516.pdf+limite+funzione+parte+intera&hl=it &gl=it&pid=bl&srcid=ADGEESjjkZp9Ce8jFTEo_T8Azz_LLv jpnkjFJWXKyf_0R1V2eGzOAaxsH85U8KlhTJqADCSdG-uwRP7qMTU87L70dG5QKwONYqemqdO8D1hJaSnnAUIM4DGAVkbT b-DIzfEv5PaYhNQk&sig=AHIEtbTJZC4WPuSWE5n0fNd6r1REynK zZQ]small paper of University of Rome[/url] (but it is in italian). So I now confirm what I've said.
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April 16th, 2012, 08:50 AM   #8
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Re: calculus doubts

Quote:
Originally Posted by Delirium
Your limit is: ; but is defined as the biggest integer smaller than ; so, if , is steadly .

But if , so I think that this limit doesn't exist, but I'm not sure.

EDIT: I found a confirmation in a www.mat.uniroma2.it/~tauraso/Online1/Lezioni/Lezioni1516.pdf+limite+funzione+parte+intera&hl=it &gl=it&pid=bl&srcid=ADGEESjjkZp9Ce8jFTEo_T8Azz_LLv jpnkjFJWXKyf_0R1V2eGzOAaxsH85U8KlhTJqADCSdG-uwRP7qMTU87L70dG5QKwONYqemqdO8D1hJaSnnAUIM4DGAVkbT b-DIzfEv5PaYhNQk&sig=AHIEtbTJZC4WPuSWE5n0fNd6r1REynK zZQ]small paper of University of Rome[/url] (but it is in italian). So I now confirm what I've said.
Alright delirium, this is what I was doing too. Thanks a ton
can u help me with question 2 as well?
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