evaluate with l'hopital's rule the limit as x goes to 0 of xcscx

Question

hartnn · Answer

@ericalaine  Hi,
$\Huge \mathcal{\text{Welcome To OpenStudy}\ddot\smile} $

you can write csc x = 1/ sin x to be able to apply  L'Hopital's rule.

anonymous · Answer

would it not be cscx/(1/x) then apply rules because that is what my book told me to do but then I'm stuck after that.....

hartnn · Answer

you can do that also but that would complicate things.
because if you differentiate 1/x, you'll get -1/x^2. 
and then you have to take that -x^2 in the denominator to get
-x^2 *(-cot^2 x)
and then if you put x=0, you get the answer.

hartnn · Answer

whereas, if you do, 
x/ sin x
after applying L'Hopital's
you directly get 1/ cos x
quite simple, right ?

anonymous · Answer

yes way more simple!

hartnn · Answer

:)
any more doubts ?

anonymous · Answer

where does sin come from exactly?

hartnn · Answer

oh, didn't you know, 
$\large \sin x=\dfrac{1}{\csc x}$

anonymous · Answer

thank you so much!

hartnn · Answer

welcome ^_^

hartnn · Answer

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