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 November 18th, 2013, 06:03 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, 06:12 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, 06:29 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, 07:49 PM #4 Global Moderator   Joined: Oct 2008 From: London, Ontario, Canada - The Forest City Posts: 7,982 Thanks: 1166 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, 08:26 PM   #5
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Re: In what situation frac of limits equals to limit of fr

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 Originally Posted by greg1313 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. If g(n) is a Cauchy sequence and we taking it as the denominator that never goes to zero, I believe the conclusion is true. Cuz the limit operation results to a real number.But infinity divides infinity, seriously? November 18th, 2013, 11:13 PM   #6
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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 (a-epsion,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 (a-epsion,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 non-negative 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(x-a)
f(x) = 1/abs(x-a) if x is rational, 2/abs(x-a) if x is irrational

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Again, inf/inf doesn't make sense regarding as a fraction, because you can only divide a real number by a real number. Therefore the equation (the first) is incorrect. November 19th, 2013, 01:48 PM   #7
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Re: In what situation frac of limits equals to limit of fr

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 Originally Posted by stainburg 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 Then you do not know what "undetermined" (or "indeterminate") means. has both numerator and denominator going to infinity so is "indeterminate". But it should be obvious, even to you, that . November 19th, 2013, 03:14 PM   #8
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Re: In what situation frac of limits equals to limit of fr

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 Originally Posted by HallsofIvy Then you do not know what "undetermined" (or "indeterminate") means. has both numerator and denominator going to infinity so is "indeterminate". But it should be obvious, even to you, that .
So, you suppose me never go to the college. What would you call infinity/infinity? November 19th, 2013, 04:00 PM   #9
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Re: In what situation frac of limits equals to limit of fr

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 Originally Posted by csak Again, inf/inf doesn't make sense regarding as a fraction, because you can only divide a real number by a real number. Therefore the equation (the first) is incorrect.
Thanks for the answer. I believe inf/inf doesn't make sense, too. Another question: how about DiracDelta(0)/DiracDelta(0) ? This truly confuses me  November 19th, 2013, 06:35 PM #10 Math Team   Joined: Jul 2011 From: North America, 42nd parallel Posts: 3,372 Thanks: 234 Re: In what situation frac of limits equals to limit of fr What if you do this ? Since  Tags equals, frac, limit, limits, situation Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post yogazen2013 Calculus 2 October 5th, 2013 09:44 AM jpanderson8 Calculus 1 September 26th, 2013 10:45 AM sivela Calculus 1 April 26th, 2010 05:53 PM oddlogic Linear Algebra 6 January 29th, 2010 09:55 AM Yegreg Number Theory 1 December 3rd, 2007 08:33 PM

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