December 31st, 2011, 12:04 AM  #1 
Member Joined: Dec 2011 Posts: 61 Thanks: 1  do this with limit
Let be a continuous function for all real values of such that the following conditions are satisfied: (I) (II) (III) for all real numbers and . Show that for all real values of . 
December 31st, 2011, 01:37 AM  #2 
Senior Member Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,211 Thanks: 521 Math Focus: Calculus/ODEs  Re: do this with limit
I'm not sure what limits have to do with this, but here is my take on it: From condition III, we must have: where r and k are nonzero real constants. From condition II, we then find: so that we have: This satisfies condition I, and we find: ' for all real x. 
December 31st, 2011, 05:02 AM  #3 
Member Joined: Dec 2011 Posts: 61 Thanks: 1  Re: do this with limit
oh ur solution is pretty cool, but give some comments with this one: 
December 31st, 2011, 10:11 AM  #4 
Senior Member Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,211 Thanks: 521 Math Focus: Calculus/ODEs  Re: do this with limit
After having some sleep, I feel kind of thick now! Of course, where derivatives are concerned there are limits. I like your solution, especially the use of L'Hôpital's rule. 

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