Saturday, 5 April 2014

multivariable calculus - Conditions to exploit Polar coordinates in limits.


Evaluate, lim(x,y)(0,0)f(x,y)=lim(x,y)(0,0)2x2yx4+y2





When I used polar coordinates with x=rcosθ,y=rsinθ,



limr0rcosθsin2θr2cos4θ+sin2θ=0



But when I use path y=x2,



lim(x,y)(0,0)2x42x4=1




Also from path x=0 or y=0 both gives,
lim(x,y)(0,0)2x2yx4+y2=0



From path knowledge, I can say Limit does not exist.



Why this occurred that I got two different values of limits from Polar and the path makes me put a question that when to employ polar coordinates method to compute limits? When can I ascertain that it gives the correct value? Why is it giving out the value 0 even when limit DNE?



Please help!

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