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January 17th, 2018, 03:37 AM  #11 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,445 Thanks: 2499 Math Focus: Mainly analysis and algebra 
Essentially your argument is one of context and the OP implies through the usr of R that we are talking about real numbers, where $0^0$ is undefined. In other contexts different definitions or conventions do apply. The example of $\displaystyle e^x=\sum_{n=0}\frac{x^n}{n!}$ is a rather amusing one. It's an expression packed with conventions and notational obfuscation. 
January 17th, 2018, 08:38 AM  #12  
Senior Member Joined: Oct 2009 Posts: 555 Thanks: 179  Quote:
Quote:
 
January 17th, 2018, 09:16 AM  #13  
Senior Member Joined: Sep 2015 From: USA Posts: 2,122 Thanks: 1102  Quote: from 1 to some big number the graph certainly tends towards 1.  
January 17th, 2018, 10:20 AM  #14 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,445 Thanks: 2499 Math Focus: Mainly analysis and algebra 
$\displaystyle \left( \frac1x \right)^{\frac1{\ln x}}$ doesn't

January 17th, 2018, 10:25 AM  #15 
Global Moderator Joined: Oct 2008 From: London, Ontario, Canada  The Forest City Posts: 7,879 Thanks: 1087 Math Focus: Elementary mathematics and beyond 
$\mathcal{O}(x)\gg\mathcal{O}(\log x)$
Last edited by greg1313; January 18th, 2018 at 06:53 AM. 
January 17th, 2018, 12:47 PM  #17 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,445 Thanks: 2499 Math Focus: Mainly analysis and algebra  Yes. More to the point $$x^{\frac1{\ln x}}=e^{\ln{x^{\frac1{\ln x}}}}=e^{\frac1{\ln x}\ln x}=e^{1}$$
Last edited by greg1313; January 18th, 2018 at 06:53 AM. 

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