January 7th, 2019, 11:14 AM  #11 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,690 Thanks: 2669 Math Focus: Mainly analysis and algebra 
How are you thinking? If the two limits are $a$ and $b$ such that $0 \ge a > b$ and $f(x) \ge a$ for all values of $x$, how does the fact that $f(x)$ is monotonically decreasing lead to a contradiction?

January 7th, 2019, 02:27 PM  #12 
Global Moderator Joined: Oct 2008 From: London, Ontario, Canada  The Forest City Posts: 7,968 Thanks: 1152 Math Focus: Elementary mathematics and beyond 
If f(x) $\ge$ a how can we say f(x) approaches b?

January 7th, 2019, 04:19 PM  #13 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,690 Thanks: 2669 Math Focus: Mainly analysis and algebra 
But that argument doesn't use the monotone property of $f(x)$, it falls back to the $\epsilon\delta$ definition of the limit, doesn't it?

January 7th, 2019, 04:49 PM  #14 
Global Moderator Joined: Oct 2008 From: London, Ontario, Canada  The Forest City Posts: 7,968 Thanks: 1152 Math Focus: Elementary mathematics and beyond  
January 7th, 2019, 07:47 PM  #15 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,690 Thanks: 2669 Math Focus: Mainly analysis and algebra 
But that sentence itself implicitly states that there is only one limit.

January 7th, 2019, 07:48 PM  #16 
Global Moderator Joined: Oct 2008 From: London, Ontario, Canada  The Forest City Posts: 7,968 Thanks: 1152 Math Focus: Elementary mathematics and beyond  
January 11th, 2019, 06:07 AM  #17 
Banned Camp Joined: Mar 2015 From: New Jersey Posts: 1,720 Thanks: 126 
Definition: $\displaystyle \lim_{x\rightarrow \infty}\frac{1}{x} = L$ if, given $\displaystyle \epsilon$, M exists such that $\displaystyle \left\frac{1}{x}L\right<\epsilon$ if $x$ > M Let $\displaystyle \epsilon = .01$ $\displaystyle \left\frac{1}{x}1\right<.01$ $\displaystyle .01<\frac{1}{x}1<.01 \rightarrow$ $\displaystyle .99<\frac{1}{x}<1.01$ There is no M such that this is true for all $x$ > M, so $L$ can't be 1. Last edited by skipjack; January 18th, 2019 at 06:20 AM. 

Tags 
limit 
Thread Tools  
Display Modes  

Similar Threads  
Thread  Thread Starter  Forum  Replies  Last Post 
upper limit = lower limit implies convergence  zylo  Calculus  13  May 31st, 2017 12:53 PM 
Limit Superior and Limit Inferior  veronicak5678  Real Analysis  4  August 22nd, 2011 10:07 AM 
Limit  unlimited  Algebra  7  April 3rd, 2011 10:14 AM 
Limit  madzia  Algebra  0  January 23rd, 2010 01:52 PM 
when should we evaluate left limit and right limit?  conjecture  Calculus  1  July 24th, 2008 01:14 PM 