April 15th, 2012, 07:35 AM  #1 
Newbie Joined: Apr 2012 Posts: 18 Thanks: 0  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(xh)}/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!) 
April 15th, 2012, 07:54 AM  #2 
Math Team Joined: Mar 2012 From: India, West Bengal Posts: 3,871 Thanks: 86 Math Focus: Number Theory  Re: calculus doubts
1. If x is integer, then x/[x] lim x > 0 is 0

April 15th, 2012, 08:18 AM  #3 
Math Team Joined: Mar 2012 From: India, West Bengal Posts: 3,871 Thanks: 86 Math Focus: Number Theory  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 
April 15th, 2012, 12:20 PM  #4  
Newbie Joined: Apr 2012 From: Padua, Italy Posts: 15 Thanks: 0  Re: calculus doubts Quote:
So  
April 15th, 2012, 12:24 PM  #5 
Newbie Joined: Apr 2012 From: Padua, Italy Posts: 15 Thanks: 0  Re: calculus doubts
You can use a quite similar argument to solve the third question...

April 15th, 2012, 08:48 PM  #6 
Newbie Joined: Apr 2012 Posts: 18 Thanks: 0  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 ?? 
April 16th, 2012, 04:38 AM  #7 
Newbie Joined: Apr 2012 From: Padua, Italy Posts: 15 Thanks: 0  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_0R1V2eGzOAaxsH85U8KlhTJqADCSdGuwRP7qMTU87L70dG5QKwONYqemqdO8D1hJaSnnAUIM4DGAVkbT bDIzfEv5PaYhNQk&sig=AHIEtbTJZC4WPuSWE5n0fNd6r1REynK zZQ]small paper of University of Rome[/url] (but it is in italian). So I now confirm what I've said. 
April 16th, 2012, 08:50 AM  #8  
Newbie Joined: Apr 2012 Posts: 18 Thanks: 0  Re: calculus doubts Quote:
can u help me with question 2 as well?  

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