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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 $\frac{\lim_{x\to a}f(x)}{\lim_{y \to a}g(y)}=\lim_{x\to a}\frac{f(x)}{g(x)}$ , $a\in \mathbb {R}$ or $\frac{\lim_{x\to \infty}f(x)}{\lim_{y \to \infty}g(y)}=\lim_{x\to \infty}\frac{f(x)}{g(x)}$
 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 $\frac{\lim_{x\to \infty}e^x}{\lim_{x\to \infty} 10^x}=\lim_{x\to \infty} \frac{e^x}{10^x}=0$. I said :"you're freakin me out, because any mathematical analysis textbook will never ever teach that. $\frac {\infty}{\infty}$ 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: 7,898 Thanks: 1093 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

Quote:
 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 $lim_{x \to a} g(x) \in \mathbb{R}$ 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

Quote:
 $\frac{\lim_{x\to \infty}e^x}{\lim_{x\to \infty} 10^x}=\lim_{x\to \infty} \frac{e^x}{10^x}=0$
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

Quote:
 Originally Posted by stainburg A Chinese physicist argued with me days ago. He believes that $\frac{\lim_{x\to \infty}e^x}{\lim_{x\to \infty} 10^x}=\lim_{x\to \infty} \frac{e^x}{10^x}=0$. I said :"you're freakin me out, because any mathematical analysis textbook will never ever teach that. $\frac {\infty}{\infty}$ 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. $\lim_{x\to\infty} \frac{x}{x}$ has both numerator and denominator going to infinity so is "indeterminate". But it should be obvious, even to you, that $\lim_{x\to\infty} \frac{x}{x}= 1$.

November 19th, 2013, 03:14 PM   #8
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Re: In what situation frac of limits equals to limit of fr

Quote:
 Originally Posted by HallsofIvy Then you do not know what "undetermined" (or "indeterminate") means. $\lim_{x\to\infty} \frac{x}{x}$ has both numerator and denominator going to infinity so is "indeterminate". But it should be obvious, even to you, that $\lim_{x\to\infty} \frac{x}{x}= 1$.
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

Quote:
 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: 233 Re: In what situation frac of limits equals to limit of fr What if you do this ? $10^x \= \ e^x \ \cdot \ \frac{10^x}{e^x}$ $\frac{ \lim_{x \to +\infty} \ e^x}{\lim_{x \to +\infty} \ 10^x } \= \ \frac{ \lim_{x \to +\infty} \ e^x}{ \lim_{x \to +\infty} \ $$e^x \cdot \frac{10^x}{e^x }$$ } \ = \ \frac{ \cancel{ \lim_{x \to +\infty} \ e^x}}{ \cancel{ \lim_{x \to +\infty} \ e^x} \ \cdot \ \lim_{x \to +\infty} \ $$\frac {10}{e}$$^x} \ = \ \frac{1}{ + \infty} \ = \ 0$ Since $\frac{ 10}{e} \ > \ 1$

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