My Math Forum  

Go Back   My Math Forum > College Math Forum > Calculus

Calculus Calculus Math Forum


Thanks Tree6Thanks
  • 2 Post By romsek
  • 1 Post By v8archie
  • 1 Post By v8archie
  • 1 Post By skipjack
  • 1 Post By SDK
Reply
 
LinkBack Thread Tools Display Modes
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:
$\displaystyle \lim_{x \to 1^-} \frac{1}{x-1} = - \infty$
$\displaystyle \lim_{x \to 1^+} \frac{1}{x-1} = \infty$
babaliaris is offline  
 
July 4th, 2019, 08:52 AM   #2
Senior Member
 
romsek's Avatar
 
Joined: Sep 2015
From: USA

Posts: 2,500
Thanks: 1372

no

$\lim \limits_{x \to 0} \dfrac{\sin(x)}{x} = 1$
Thanks from topsquark and babaliaris
romsek is online now  
July 4th, 2019, 09:11 AM   #3
Math Team
 
Joined: Dec 2013
From: Colombia

Posts: 7,671
Thanks: 2651

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
v8archie is offline  
July 4th, 2019, 11:46 AM   #4
Math Team
 
Joined: Dec 2013
From: Colombia

Posts: 7,671
Thanks: 2651

Math Focus: Mainly analysis and algebra
Quote:
Originally Posted by babaliaris View Post
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.
Thanks from babaliaris
v8archie is offline  
July 4th, 2019, 09:31 PM   #5
Global Moderator
 
Joined: Dec 2006

Posts: 20,823
Thanks: 2160

Quote:
Originally Posted by babaliaris View Post
. . . can not be simplified any more
How is that property defined?
Thanks from SDK
skipjack is offline  
July 5th, 2019, 12:04 PM   #6
Senior Member
 
Joined: Oct 2015
From: Greece

Posts: 137
Thanks: 8

Quote:
Originally Posted by skipjack View Post
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.
babaliaris is offline  
July 6th, 2019, 01:45 PM   #7
SDK
Senior Member
 
Joined: Sep 2016
From: USA

Posts: 635
Thanks: 401

Math Focus: Dynamical systems, analytic function theory, numerics
Quote:
Originally Posted by babaliaris View Post
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.
Thanks from babaliaris

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

  My Math Forum > College Math Forum > Calculus

Tags
fx or qx, limit, simplified



Thread Tools
Display Modes


Similar Threads
Thread Thread Starter Forum Replies Last Post
Simplified expression life24 Pre-Calculus 3 January 15th, 2017 11:30 AM
Can this be simplified? iScience Algebra 1 August 31st, 2016 12:56 PM
Which of these would be the most simplified? Dalekcaan1963 Algebra 2 December 12th, 2015 08:57 PM
Can this be simplified any further? mms Algebra 7 November 2nd, 2011 08:08 AM
can this be simplified for S? empiricus Algebra 13 April 27th, 2010 06:55 PM





Copyright © 2019 My Math Forum. All rights reserved.