January 5th, 2019, 05:26 AM  #1 
Member Joined: Apr 2017 From: India Posts: 64 Thanks: 0  limit
Using delta epsilon definition, please give me a proof of how limit x tends to infinity of the function f(x) = 1/x is not equal to 1. I am confused in the choice of delta. 
January 5th, 2019, 01:14 PM  #2 
Global Moderator Joined: May 2007 Posts: 6,762 Thanks: 697 
You can do the mechanics. $\lim_{x\to \infty} \frac{1}{x}1=1\ne 0$.

January 5th, 2019, 05:33 PM  #3  
Math Team Joined: Dec 2013 From: Colombia Posts: 7,660 Thanks: 2635 Math Focus: Mainly analysis and algebra 
The definition of the limit says that $$\lim_{x \to \infty} \frac1x = 1$$ means that Quote:
I suggest that you consider some value of $\epsilon < 1$  for example $\epsilon = \frac12$. Last edited by v8archie; January 5th, 2019 at 05:41 PM.  
January 6th, 2019, 06:38 AM  #4 
Senior Member Joined: May 2016 From: USA Posts: 1,310 Thanks: 551 
Alternatively, you could prove that the limit is something other than 1.

January 6th, 2019, 06:42 AM  #5 
Senior Member Joined: Oct 2009 Posts: 784 Thanks: 280  
January 6th, 2019, 09:10 AM  #6 
Senior Member Joined: May 2016 From: USA Posts: 1,310 Thanks: 551  
January 6th, 2019, 04:06 PM  #7 
Newbie Joined: Nov 2013 Posts: 28 Thanks: 8  
January 6th, 2019, 07:26 PM  #8 
Senior Member Joined: Oct 2009 Posts: 784 Thanks: 280  
January 7th, 2019, 10:42 AM  #9 
Senior Member Joined: May 2016 From: USA Posts: 1,310 Thanks: 551  
January 7th, 2019, 10:57 AM  #10 
Global Moderator Joined: Oct 2008 From: London, Ontario, Canada  The Forest City Posts: 7,935 Thanks: 1129 Math Focus: Elementary mathematics and beyond 
The derivative of f(x) = 1/x, 1/x$^2$, tells us f(x) is monotone decreasing. Is that sufficient to tell us the limit is unique (if it exits)? I think it is.


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