Calculate the following limits, if they exist, by using a combination of polar c

glasskerfu 2021-10-22 Answered
Calculate the following limits, if they exist, by using a combination of polar coordinates and de L’Hopital rule.
lim(x,y)(0,0)arctan(x2+y2)x2+y2
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l1koV
Answered 2021-10-23 Author has 100 answers
changing into polar coordinates
x=rcosθ
y=rsinθ
x2+y2=r2cos2θ+r2sin2θ
=r2(cos2θ+sin2θ)
=r2(cos2θ+sin2θ=1)
when (x,y)(0,0)
r0
Now lim(x,y)(0,0)arctan(x2+y2)x2+y2
=limr0arctan(r2)r2
=limr011+(r2)22r2r (applying L'Hopital rule)
=limr011+r4
=1
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Jeffrey Jordon
Answered 2022-06-26 Author has 2313 answers

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