determine whether lim exists

for f(x,y) = x^4/(x^2+y^2)
as (x,y)- (0,0)

Question

determine whether lim exists 

for f(x,y) = x^4/(x^2+y^2)
as (x,y)-> (0,0)

anonymous · Answer

Have you tried converting to polar?

anonymous · Answer

no i haven't is it necessary ?

anonymous · Answer

Might not be, but that's my first impression. Converting the polar changes the denominator to $r^2$ and the numerator to $r^4\cos^4\theta$. Then as $(x,y)\to(0,0)$, I believe it would be $(r,\theta)\to(0,0)$.

anonymous · Answer

polars i tend to forget but i see what you mean,

anonymous · Answer

Wolfram says the limit turns out to be 0, which works well with my suggestion.

anonymous · Answer

i did it too on wolfram jus wanted more depth into how, can also you replace y=x

anonymous · Answer

making it into x^4/2x^2 giving you zero

anonymous · Answer

Yes, you could do that, but there's no guarantee that the result of that will give you the right answer. You have to consider "all" possible paths as $(x,y)\to(0,0)$. For example (not regarding this limit, but in general), substituting $y=x$ may give you 0, but substituting $x=0$ might not, and so on. Which is why I prefer the polar conversion method.

anonymous · Answer

okay thanks ill keep that in mind

anonymous · Answer

You're welcome.