September 24th, 2011, 01:53 PM  #1 
Member Joined: Sep 2011 Posts: 44 Thanks: 0  Limit statements
which of the following statements MUST be true, MIGHT be true, or is NEVER true about a function f which is defined for all real numbers? Justify your answer. correct me if I am wrong. a) lim x>a f(x)=f(a) might be true if this is a continuous function b)lim x>0 f(x)/x=1 then f(0)=0 might because??? c)If lim x> f(x)/x=1 then lim x>0 f(x)=0 must be true because the hole is at 0, so f(x)=0 
September 24th, 2011, 03:00 PM  #2  
Global Moderator Joined: May 2007 Posts: 6,807 Thanks: 717  Re: Limit statements Quote:
b) if f(0) not 0 (assuming continuity), then the limit is infinite, so f(0)=0 is necessary, although not sufficient. c) seems to have a typo: x > 0? question seems to be a slight variation of b).  
September 24th, 2011, 04:03 PM  #3 
Member Joined: Sep 2011 Posts: 44 Thanks: 0  Re: Limit statements
c)If lim x>0 f(x)/x=1 then lim x>0 f(x)=0

September 25th, 2011, 05:24 AM  #4 
Global Moderator Joined: Nov 2009 From: Northwest Arkansas Posts: 2,766 Thanks: 4  Re: Limit statements
b) might c) must Isn't saying "... Is necessary" the same as saying "must"? 
September 25th, 2011, 02:04 PM  #5 
Global Moderator Joined: May 2007 Posts: 6,807 Thanks: 717  Re: Limit statements
For b) must. I don't see any real difference between b) and c).

September 25th, 2011, 02:59 PM  #6 
Global Moderator Joined: Nov 2009 From: Northwest Arkansas Posts: 2,766 Thanks: 4  Re: Limit statements
What about f(x) = 7 when x = 0, = x otherwise? The limit as x  > a (for all real a) is 1... 
September 26th, 2011, 10:00 AM  #7  
Global Moderator Joined: Nov 2009 From: Northwest Arkansas Posts: 2,766 Thanks: 4  Re: Limit statements Quote:
c) refers to the value of the function f at x = 0 That b) isn't "must" can also be seen by considering f(x) = e^x. Then f(x)/x = e^x/x, which has limit of 1 as x approaches 0 (by L'Hospital). Yet f(0) = 1.  
September 26th, 2011, 12:50 PM  #8  
Global Moderator Joined: May 2007 Posts: 6,807 Thanks: 717  Re: Limit statements Quote:
 
September 26th, 2011, 01:14 PM  #9 
Global Moderator Joined: Nov 2009 From: Northwest Arkansas Posts: 2,766 Thanks: 4  Re: Limit statements
Ah I got carried away! What about x^2/x ? 

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