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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: $\displaystyle \lim_{x \to 1} \frac{1}{x1}$ diverges because: $\displaystyle \lim_{x \to 1^} \frac{1}{x1} =  \infty$ $\displaystyle \lim_{x \to 1^+} \frac{1}{x1} = \infty$ 
July 4th, 2019, 08:52 AM  #2 
Senior Member Joined: Sep 2015 From: USA Posts: 2,554 Thanks: 1403 
no $\lim \limits_{x \to 0} \dfrac{\sin(x)}{x} = 1$ 
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 onesided limits. (e.g. $\frac1{x^2}$).

July 4th, 2019, 11:46 AM  #4  
Math Team Joined: Dec 2013 From: Colombia Posts: 7,685 Thanks: 2666 Math Focus: Mainly analysis and algebra  Quote:
$\displaystyle \lim_{x \to 1^} \frac{1}{x1} = \infty$ 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 
Global Moderator Joined: Dec 2006 Posts: 20,978 Thanks: 2229  
July 5th, 2019, 12:04 PM  #6 
Senior Member Joined: Oct 2015 From: Greece Posts: 137 Thanks: 8  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 
Senior Member Joined: Sep 2016 From: USA Posts: 647 Thanks: 412 Math Focus: Dynamical systems, analytic function theory, numerics  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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fx or qx, limit, simplified 
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