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

Cheyanne Leigh 2021-10-14 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)1cos(x2+y2)x2+y2
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falhiblesw
Answered 2021-10-15 Author has 97 answers

Given: The function is lim(x,y)(0,0)1cos(x2+y2)x2+y2
Consider x=rcosθ, and y=rsinθ.
The expression becomes limr01cos(r2)r2
Conclusion:
Apply L’Hopital rule as follows:
limr01cos(r2)r2=limr0(sin(r2)2r)2r
=limr0sin(r2)
=sin(0)
=0
Hence, lim(x,y)(0,0)1cos(x2+y2)x2+y2=0

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Jeffrey Jordon
Answered 2022-06-26 Author has 2313 answers

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