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November 18th, 2013, 05:03 PM  #1 
Senior Member Joined: Oct 2010 From: Changchun, China Posts: 492 Thanks: 14  In what situation frac of limits equals to limit of frac?
I mean , or 
November 18th, 2013, 05:12 PM  #2 
Senior Member Joined: Oct 2010 From: Changchun, China Posts: 492 Thanks: 14  Re: In what situation frac of limits equals to limit of fr
Of course, we have to suppose that limit of g(x) exists when x goes to a rational number 'a' or infinity but it never goes to zero! 
November 18th, 2013, 05:29 PM  #3 
Senior Member Joined: Oct 2010 From: Changchun, China Posts: 492 Thanks: 14  Re: In what situation frac of limits equals to limit of fr
A Chinese physicist argued with me days ago. He believes that . I said :"you're freakin me out, because any mathematical analysis textbook will never ever teach that. is indetermined, and you can never tell how fast the x's in numerator and denominator go to infinity!" Of course, as he said, I go to hell 
November 18th, 2013, 06:49 PM  #4 
Global Moderator Joined: Oct 2008 From: London, Ontario, Canada  The Forest City Posts: 7,923 Thanks: 1123 Math Focus: Elementary mathematics and beyond  Re: In what situation frac of limits equals to limit of fr The limit laws aren't much help in determining the limit but I believe what he says is correct. I'm sure there is a warm spot for me down there as well. 
November 18th, 2013, 07:26 PM  #5  
Senior Member Joined: Oct 2010 From: Changchun, China Posts: 492 Thanks: 14  Re: In what situation frac of limits equals to limit of fr Quote:
 
November 18th, 2013, 10:13 PM  #6  
Senior Member Joined: Feb 2013 Posts: 281 Thanks: 0  Re: In what situation frac of limits equals to limit of fr
f(x): R > R g(x): R > R 1) Suppose there is an epsilon such that g(x)<>0 if x in (aepsion,a+epsilon) interval and and lim g(x)<>0. Then f converges to a real number <> lim f / lim g = lim f/g. f diverges to +inf <> f/g diverges to +inf f divergent <> f/g divergent 2) Suppose there is an epsilon such that g(x)<>0 if x in (aepsion,a+epsilon) interval and lim g(x)=0. Then lim f/g can exist and can not exist, can converge to any real number or to +/inf. lim f / lim g is undefined, becasuse divived by zero. 3) Suppose lim f = lim g = + inf. Then lim f/g can exist or can not exist. Lim f/g can converge to +inf or any nonnegative real, but not to negative real. lim f / lim g is undefined, becasuse division is defined (by definition) on real numbers. An example when lim f/g doesn't exist: g(x) = 1/abs(xa) f(x) = 1/abs(xa) if x is rational, 2/abs(xa) if x is irrational Quote:  
November 19th, 2013, 12:48 PM  #7  
Math Team Joined: Sep 2007 Posts: 2,409 Thanks: 6  Re: In what situation frac of limits equals to limit of fr Quote:
 
November 19th, 2013, 02:14 PM  #8  
Senior Member Joined: Oct 2010 From: Changchun, China Posts: 492 Thanks: 14  Re: In what situation frac of limits equals to limit of fr Quote:
 
November 19th, 2013, 03:00 PM  #9  
Senior Member Joined: Oct 2010 From: Changchun, China Posts: 492 Thanks: 14  Re: In what situation frac of limits equals to limit of fr Quote:
 
November 19th, 2013, 05:35 PM  #10 
Math Team Joined: Jul 2011 From: North America, 42nd parallel Posts: 3,372 Thanks: 233  Re: In what situation frac of limits equals to limit of fr
What if you do this ? Since 

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