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anonymous
 one year ago
Please help find the limit (2 Variables)
lim(x,y) —> (0,0)
of
tan(x^2 +y^2)arctan(1/(x^2 +y^2))
anonymous
 one year ago
Please help find the limit (2 Variables) lim(x,y) —> (0,0) of tan(x^2 +y^2)arctan(1/(x^2 +y^2))

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IrishBoy123
 one year ago
Best ResponseYou've already chosen the best response.2have you tried just plugging in x,y = 0?

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0yes. but you will end up with tan(0)arctan(1/0)

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0can we use Lhopitals rule inside the arctan function

IrishBoy123
 one year ago
Best ResponseYou've already chosen the best response.2what is tan \((\pi /2) \) ??

IrishBoy123
 one year ago
Best ResponseYou've already chosen the best response.2dw:1443254278447:dw tan pi/2 is undefined and then that behavious repeats itself meaning that arctan(1/0) is defined so you have 0 x something finite = 0

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0what is the value of arctan(1/0) ?

IrishBoy123
 one year ago
Best ResponseYou've already chosen the best response.2\[(2n+1) \dfrac{\pi}{2}\]

IrishBoy123
 one year ago
Best ResponseYou've already chosen the best response.2dw:1443254619111:dw

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0so the answer is 0, but how do you obtain the value of arctan(1/0)

IrishBoy123
 one year ago
Best ResponseYou've already chosen the best response.2dw:1443254723319:dw you have \[0 \times (2n+1)\dfrac{\pi}{2} = 0\]

IrishBoy123
 one year ago
Best ResponseYou've already chosen the best response.2if that's to brutal http://math.stackexchange.com/questions/711194/howtoprovethatlimitofarctanxasxtendstoinfinityispi2

IrishBoy123
 one year ago
Best ResponseYou've already chosen the best response.2if you want some formalism, this makes sense, but i think it might be OTT: \(lim_{x,y \to 0,0} \, \tan(x^2 +y^2).\arctan(\dfrac{1}{x^2 +y^2})\) let \(z = x^2 + y^2\) \(lim_{x,y \to 0,0} \, z = 0\) \(\implies lim_{z \to 0} \tan z . arctan \dfrac{1}{z}\) \(= lim_{z \to 0} \tan z \times \,lim_{z \to 0} \arctan \dfrac{1}{z}\) let \(w = \dfrac{1}{z}\) \(\implies lim_{z \to 0} \tan z \times \,lim_{w \to +\infty} \arctan w\) \( lim_{z \to 0} \tan z = 0\) \(\lim_{w\to\frac\pi 2}\tan w=+\infty\iff \lim_{w\to+\infty}\arctan w=\frac\pi2\) \(\therefore lim_{x,y \to 0,0} \, \tan(x^2 +y^2).\arctan(\dfrac{1}{x^2 +y^2}) = 0\)
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