Please help to find the limit with hopital's rule

[mmmladenov1]

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[blamocur]

ég tala ekki þýsku

 

[mmmladenov1]

ég tala ekki þýsku

You can ignore the first one, the rest is as mentioned in the title of the thread.

 

[Dr.Peterson]

View attachment 29642,,,,,,,,View attachment 29643View attachment 29644
View attachment 29645

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:

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[JeffM]

I am fuzzy on what is going on after n $\ge$ 1, but I think this starts

It can be proved by complete induction that:

$$f(x) = x \cdot sin(x) \implies \text {the n}^{th} \text { derivative} = x \cdot sin \left (x + \dfrac{n \pi}{2} \right ) + n \cdot sin \left ( x + \dfrac{(n - 1) \pi}{2} \right ).$$
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.