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December 31st, 2011, 12:04 AM   #1
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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 .
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December 31st, 2011, 01:37 AM   #2
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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 non-zero real constants.



From condition II, we then find:



so that we have:



This satisfies condition I, and we find:
'
for all real x.
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December 31st, 2011, 05:02 AM   #3
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Re: do this with limit

oh ur solution is pretty cool, but give some comments with this one:

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December 31st, 2011, 10:11 AM   #4
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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'H˘pital's rule.
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