Please help to find the limit with hopital's rule

Dr.Peterson

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Nov 12, 2017
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I can't even tell how many separate questions there are here (there are two "part b"s!), or what the instructions apply to.

Please make some attempt to follow the rules:
 

JeffM

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Sep 14, 2012
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I am fuzzy on what is going on after n [imath]\ge[/imath] 1, but I think this starts

It can be proved by complete induction that:

[math]f(x) = x \cdot sin(x) \implies \text {the n}^{th} \text { derivative} = x \cdot sin \left (x + \dfrac{n \pi}{2} \right ) + n \cdot sin \left ( x + \dfrac{(n - 1) \pi}{2} \right ).[/math]
I am guessing that “complete induction” is what we call “weak mathematical induction.” I suspect this theorem is to be used (rather than proved) to find the indicated limits.
 
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