My Math Forum Limit of f(x)/q(x) with q(c) = 0 that can not be simplified.
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 July 4th, 2019, 05:09 AM #1 Senior Member   Joined: Oct 2015 From: Greece Posts: 137 Thanks: 8 Limit of f(x)/q(x) with q(c) = 0 that can not be simplified. Is it true that if $\displaystyle \lim_{x \to c} \frac{f(x)}{q(x)}$ with q(c) = 0, $\displaystyle f(c) \in R$ and if that fraction can not be simplified any more, then this limit will always diverge. Example: $\displaystyle \lim_{x \to 1} \frac{1}{x-1}$ diverges because: $\displaystyle \lim_{x \to 1^-} \frac{1}{x-1} = - \infty$ $\displaystyle \lim_{x \to 1^+} \frac{1}{x-1} = \infty$
 July 4th, 2019, 08:52 AM #2 Senior Member     Joined: Sep 2015 From: USA Posts: 2,554 Thanks: 1403 no $\lim \limits_{x \to 0} \dfrac{\sin(x)}{x} = 1$ Thanks from topsquark and babaliaris
 July 4th, 2019, 09:11 AM #3 Math Team   Joined: Dec 2013 From: Colombia Posts: 7,685 Thanks: 2666 Math Focus: Mainly analysis and algebra On the other hand, if $f(x)$ has a Taylor series $T(f, c)$ about $x=c$ and $q(x)$ has a Taylor series $T(q, c)$ about $x=c$ and $\frac{T(f,c)}{T(q,c)}$ cannot be simplified, I think that the limit won't exist, but you won't always have different one-sided limits. (e.g. $\frac1{x^2}$). Thanks from topsquark
July 4th, 2019, 11:46 AM   #4
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Quote:
 Originally Posted by babaliaris Example: $\displaystyle \lim_{x \to 1} \frac{1}{x-1}$ diverges because: $\displaystyle \lim_{x \to 1^-} \frac{1}{x-1} = - \infty$ $\displaystyle \lim_{x \to 1^+} \frac{1}{x-1} = \infty$
$\frac1{x-1}$ doesn't converge to a limit as $x\to1$ but not because the one-sided limits differ. It's because they don't exist.

$\displaystyle \lim_{x \to 1^-} \frac{1}{x-1} = -\infty$ doesn't mean that there is a limit with a value of $-\infty$. It's a shorthand for saying that the function grows without bound in the negative direction - which means that the limit doesn't exist.

July 4th, 2019, 09:31 PM   #5
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Quote:
 Originally Posted by babaliaris . . . can not be simplified any more
How is that property defined?

July 5th, 2019, 12:04 PM   #6
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Quote:
 Originally Posted by skipjack How is that property defined?
I wasn't trying to specify a rule, just an observation. "Can not be simplified any more", it wasn't my intention to specify this as a property.

Last edited by skipjack; July 7th, 2019 at 12:59 AM.

July 6th, 2019, 01:45 PM   #7
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Quote:
 Originally Posted by babaliaris I wasn't trying to specify a rule, just an observation. "Can not be simplified any more", it wasn't my intention to specify this as a property.
Skipjack's point is that the phrase "can not be simplified any more" does not have any meaning. So nobody knows how to answer your question. This would be like me asking you to solve $3y + 4x = 7$ when $x$ is green. It is meaningless.

Last edited by skipjack; July 7th, 2019 at 12:59 AM.

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