Given $$f(x,y) = \frac {2x^4-5x^2y^2 + y^5}{(x^2 + y^2)^2} , (x,y) \neq (0,0) ~~\mbox{and} ~~f(x,y) = 0 , (x,y) =(0,0)$$

Find a $\delta$ such that $|f(x,y)-f(0,0)| < 0.01$ whenever $\sqrt {x^2+ y^2} < \delta $

i tried to go into polar coordinates but unable to find it.


There is no such $\delta$. Your function is not continuous at $(0,0)$, (consider what happens at $f(x,0)$ as $x$ approaches $0$.)

  • $\begingroup$ i tried to go polar corodinate. i saw that there is trig function which hasnot any r attached to it. so limit doesnot exists. maybe question is wrong $\endgroup$ – Gathdi Jun 12 at 3:35
  • $\begingroup$ @Gathdi $\lim_{x\to 0} f(x,0)=2$ so $f(x,0)$ can not get close to $f(0,0)=0$. $\endgroup$ – Julian Mejia Jun 12 at 3:41

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